TAME ALGEBRAS
(ON ALGORITHMS FOR SOLVING VECTOR SPACE PROBLEMS. II)
Claus Michael Ringel
Table of contents
Introduction
.I
.2
.3
.4
.5
.6
.7
The tame one-relation algebras
Quivers
Quivers with relations
Splitting zero relations
The representation types
The classification of the tame one-relation algebras
Outline of proof
The combinatorial part of the proof
page
143
143
145
150
155
157
172
177
2.
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
18,1
Concealments, finite enlargements and glueing
181
The Auslander-Reiten quiver
185
Quivers without oriented cycles
191
Concealed quiver algebras
199
Vectorspace categories
Subspace categories arising from simple injective modules 206
212
Finite enlargements
215
Glueing of two components
222
Glueing a la MHller (~)
225
Glueing using splitting zero relations
3.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Regular enlargements
Pattern
The regular modules of a tame quiver
Calculations of pattern
Some non-domestic pattern
Regular modules M with Hom(M,M) tame
Proof of theorem 3
Some components of the Auslander-Reiten quiver
Further examples of non-domestic tame algebras
Two special one-relation algebras
References
229
229
230
233
241
251
259
263
271
279
284
138
Aim of these notes is to report on some of the main algorithms
developed
recently for solving vectorspace
problems,
the most impres-
sing ones being due to the Kiev school of Nazarova and Roiter,
presents part of lectures
theory at Ottawa,
;979.
choosen one particular
difficulties,
it
~) held at the workshop on representation
For the presentation
of these methods, we have
question which can be handled without
namely the determination
This question was considered
too many
of all tame one-relation
lately by Shkabara
[36]
algebras.
and Zavadskij
[39], and we would like to outline a proof of their results which illustrates some of the recent techniques:
on the one hand, the use of par-
tially ordered sets and their representations,
vectorspace
categories
and their subspaces,
or, more generally,
and, on the other hand,
that of irreducible maps, or the global Auslander-Reiten
These two techniques usually are considered
affinity either to Kiev or Boston.
sets and vectorspace
Auslander-Reiten
classes.
indicating an
they actually fit together
Namely,
partially ordered
categories will be derived directly from certain
quivers.
be of independent
quiver.
separately,
However,
very well, as we would like to demonstrate.
of
The pattern which appear in this way seem to
interest;
they fall into a small number of similarity
Some of them may be indexed by the extended Dynkin diagrams,
they represent
typical
"non-domestic"
tame vectorspace
problems.
This
is a futher objective of these lectures:
we would like to spread some
better understanding
of tame situations.
Recall that an algebra is
called tame provided
there are at most one-parameter
composable modules,
so that it is possible
fication of all indecomposable
modules.
families of inde-
to obtain a complete classi-
We will call a tame algebra
domestic in case there is a finite number of one-parameter
such that all other one-parameter
modules
from
familes are obtained by extending
F
by themselves
(for a precise definition,
The non-domestic
tame algebras
seem to be of particular
as we will see, they have some surprising properties.
of non-domestic
tame algebras,
tures will be associated
In classifying
establish
tame.
families
however,
see
1.4).
interest,
and
There are plenty
those considered
in these lec-
to very few similarity classes of pattern.
tame algebras the usually difficult part is to
that those algebras which are claimed to be tame actually are
There are three steps of insight:
~) The first part dealt with the Brauer-Thrall conjectures, see
The written texts of these two parts are mutually independent.
[35].
139
(i)
the first is to determine just the representation
out giving a complete
situations,
list of all the indecomposable
in most of the non-domestic
type, with-
modules.
In many
ones, this is at present
the
only feasible goal;
(ii)
modules,
the next step is to give a list of all the indecomposable
and perhaps even an algorithm for deciding whether a given mo-
dule is isomorphic
which decomposes
to one of the modules
a given module into a direct sum of indecomposable
dules from the list.
complete
0nly for very few vectorspace
list of indecomposable
(iii)
modules,
in the list, or an algorithm
modules
problems
mo-
such a
is known;
the final step is to describe completely the category of
not only the indecomposable
first approximation
the Auslander-Reiten
modules but also all the maps.
As
one would like to know all irreducible maps, thus
quiver of the category.
Note that the Kiev school seems to be concentrated
on the first
goal, whereas Auslander usually stresses the third aspect of considering
maps.
In dealing with the tame one-relation
gather as much information as possible.
algebras, we will try to
Now, for the domestic algebras
it will be easy to establish not only the representation
to determine
all indecomposable
modules,
third step aiming at a description
is the goal of part
2
and section
discuss four constructions
starting with tame quivers:
of components
type, but also
thus one should proceed to the
of the Auslander-Reiten
3.7
concealments,
finite enlargements,
and certain regular enlargements.
quiver of an algebra is not only of interest
that it can be used for the consideration
glueing
In all cases it is not
the new Auslander-Reiten
quiver.
Let us stress again the fact that the determination
part
This
we will
which can be used to built domestic algebras
difficult actually to determine
der-Reiten
quiver.
of these lectures:
of the Auslanin itself, but
of further enlargements.
3, this will be done, we will consider regular enlargements
In
not
only of tame quivers but also of algebras obtained before,
and, in this
way, many non-domestic
In the case
of non-domestic
tame algebras will be constructed.
algebras, we will content ourselves with establishing
the representation
type without considering
further steps
(ii)
or
(iii).
The problem of determining
some classes of tame algebras has attrac-
140
ted a lot of attention,
Zavadskij,
lately.
Besides,
we also should mention Marmaridis
of the result of Shkabara using different
Loupias
hand,
the work of Shkabara and
[23].
[24]
who obtained part
techniques,
namely those of
Note that our approach is rather similar.
S. Brenner also has considered many non-domestic
[ 8 ], using a generalization
tion functors developed
in her joint work with Butler
which we usually will neglect.
ding vectorspace
a n d Zavadskij
quadratic
[ 9 ].
She con-
Also, there is a recent paper by
(of type
(~4' 2 • 2), see
of two non-
3.5) to a eorrespon-
*)
problem
let us confess that our interest
in the work of Shkabara
was motivated by the fact that the report at the confe-
rence was intended to include the theory of differential
gories due to Kleiner and Roiter.
and in particular,
graded cate-
in order to get a better understan-
ding of this method it seemed to be convenient
applications,
reflec-
form, an aspect
[14], which reduces the investigation
domestic tame algebras
Finally,
tame algebras
of the Bernstein-Gelfand-Ponomarev
siders with any algebra the corresponding
Donovan-Freislich
On the other
to follow its recent
to see at what point the previously
known methods were not strong enough for solving the problems.
ferential
graded categories were introduced
as generalisation
The difof the
method of partially ordered sets, and, in fact, both Shkabara and Zavadskij need a further generalisation,
gories.
However,
tion algebras,
namely differential ~-graded
a rather straight
forward reduction immediately
to a subspace problem of a vectorspace
partially
ordered set.
Also,
category,
considered
by Shkabara)
In fact,
relation algebras,
leads
and usually even to a
in this way, we see that there is no in-
trinsic difference between quivers with a commutativity
by Zavadskij).
cate-
it turned out that, starting with most tame one-rela-
relation
and quivers with a zero relation
(as
(as considered
in the same manner one can deal with all one-
and even with many quivers with more relations.
These notes are organized as follows:
general techiques which will be used:
cluding the additive categories
there are reports on the two
the vectorspace
categories
of partially ordered sets
in-
(2.4), and
~) Note that in contrast to a claim in the paper, Donovan-Freislich do
not give a classification of the indecomposable modules, they determine
only the representation type. There is a list of dimension types of
indecomposable modules in the paper, which however is incomplete.
I4I
the Auslander-Reiten
quivers
theory of tame quivers.
(2.1).
Also, we frequently will need the
The tame quivers and their representations
been classified by Donovan-Freislich
[12]
and Nazarova
have
[26], however,
use will be made of the full structure theory of the corresponding module category,
as established
there is a report on it in
in the joint work with Dlab
(2.2)
and
(3.2).
quivers without or with relations will be found in
the definition
remaining
of the various representation
and
(1.2),
(1.4).
The
In order to stress
we give proofs even of some very elementary
as the splitting zero relations
included on many examples.
of the tame one-relation
one-relation
(I.I)
types is in
sections are rather self-contained.
various techniques,
[II], and
The main notions on
algebras
in
(1.3), and detailed
In part
algebras
(theorem
(theorem
2).
i)
and the minimal wild
Also it is shown that the listed
arguments
are provided which are needed to prove that every one-relation
or specializes
ted.
Part
arguments
from quivers.
full Auslander-Reiten
and, in particular,
Theorem
space categories
r
3
listed in theorem
from Shkabara and Zavadskij
2.
Part
3
However,
for obtaining domes-
we obtain a large amount of non-domestic
corresponding
a regular F-module,
The one-relation
with
type, and
algebras
they turn out to provide also
Namely, we will show that the non-domestic
which we encounter have rather strange properties
certain one-relation
tame alge-
the vector-
to their representation
similarity classes.
are not only our object of investigation;
a method of proof.
it classifies
Hom(Mp,Mp) , M F
according
the
consider~ regular enlargements
seems to he of interest:
of the form
1
are not repea-
Here, our aim is always to determine
quiver.
a tame connected quiver,
determines
algebra
listed in theorem
2 deals with some special constructions
tic algebras
bras.
of one of the algebras
to one of the algebras
the combinatorial
is
I, we present the classification
wild algebras actually are wild and some of the combinatorial
is either a specialization
facts,
information
(see
pattern
3.4), using
algebras.
We assume throughout the paper that our base field k is algebra*)
ically closed
• All modules will be assumed to be finitely generated,
and usually will be right modules.
called finite provided
it contains
An additive
category of modules
is
only a finite number of indecompo-
*) Most of the result can be adapted to the case of an arbitrary (commutative) base field. Note that in contrast to a remark in [14], the
extension to skew fields will provide substantial changes, since for a
skew field D, the polynomial ring D[T]
in one variable may be wild.
142
sable modules.
A full subcategory is called cofinite provided there
are only finitely many isomorphism classes of indecomposable objects
which are not in the subcategory.
We are endebted to S. Brenner and V. Dlab for several discussions
concerning quivers with commutativity relations.
Also, I would like to
express my thanks to Mrs. FettkSther and Mrs. Oberschelp for their careful and patient typing of this manuscript.
Finally, I have to thank
D. Happel, and M. H~nsch, A. HSwelmann, H. Klages, N. Kuberski,
P.R. Kurth,
W. Meier, R. MHller, P. SudhSlter, L. Unger, D. Vossieck for spotting many
misprints and inaccura ies in a first draft of these notes.
143
I.
THE TAME ONE-RELATION
ALGEBRAS
As we have m e n t ~ n e d in the introduction,
of the main techniques
of present-day
we want to report on some
representation
line of giving an outline of the classification
tion algebras.
nate
Also, we would like, at the same time, to dissemi-
some feeling for "tame" situations.
concentrated
on in parts
the classification
2
and
3:
here,
of the tame one-relation
with the easy part of the proof:
ones which may be tame.
tions.
theory along the
of the tame one-rela-
This last goal will be
in part
I, we will state
algebras,
and we will deal
that the listed algebras are the only
Let us start by recalling
the basic defini-
In this way, we also will fix the notation which will be used
in the sequel.
I.l.
Quivers
A quiver
a set
FI
F = (Po,FI)
of "arrows"
of "vertices"
and
such that to any arrow, there is assigned
is given by a set
its
starting point and its endpoint
coincide).
For example,
Fo
(these are two vertices which may
this is a typical quiver:
(Note that we usually will draw the vertices of a quiver as small
circles,
in contrast
to the elements of partially ordered sets which
we also have to draw rather frequently,
points).
duced by Gabriel
[17]
in order to formulate
problems rather efficiently,
of the representation
of
F
over
k
If
is of the form
~
over
= ~ h i-
r
of the form
linear transformation
k
is a field, a representation
(Vi,~)
the linear transformation
is, by definition,
certain vectorspace
where,
Vi, and for any arrow
are representations
of
was intro-
and it dominates now a rather large part
theory,
we have the vectorspaee
nj~
and which will be drawn as
The notion of a quiver and its representations
: V i --+ Vj.
If
k, then a map
for any vertex
i C ro,
9
~
>9, we have
l
j
(Vi,~ ~) and
(V~,~)
~ : (Vi,~ ~)
n = (hi), where
÷ (VI,~ ~)
H i : V i ÷ V~l
o ~ >9, we have
i
J
In this way, the representations of F over
is a
such that for any
abelian category, which we will denote by
MF
or
k
form an
MkF (it is the
144
module category over the path algebra
called in the next section).
kF, its definition will be re-
In particular, we usually are interested
in the indecomposable representations.
In
[17]
Gabriel has shown
that a quiver has only a finite number of indecomposable representations if and only if it is the disjoint union of quivers of the form
A
o-o-o
...
o-o--o
Dn
o-o--o
...
o ~
E6
o--o-~-o--o
E7
o--o-~-o-o
E8
o-o~-o-o--o--o
n
with arbitrary orientations of the edges.
ly many indecomposable representations,
Some quivers with infinite-
the socalled extended Dynkin
diagrams,
0
o <°
have attracted much interest
([12],[26]):
they are the only connected
quivers of tame representation type, and are even domestic (for the
definition of the possible representation types see
1.4).
The re-
presentation theory for any of these quivers with the exception of
~n' does not depend on the orientation of the edges;
pends on the number of arrows in one direction.
quiver with
p
arrows in clockwise direction and
ter clockwise direction, where
p+q = n+1, is
j,o
~2
for
~n' it de-
The prototype of the
q
arrows in coun-
145
The representation
theory of the tame quivers will play a dominant role
in the further investigations,
If
V = (Vi,~)
with vertices
~n, with
it will be recalled in
is a representation
{1,...,n}
its dimension
of
type
P
2.2
where
dim V
and
P
3.2.
is a quiver
is an element of
(dim V) i = dim V i.
If there exists a unique indecomposable
representation
sion type
x E ~n, we often will use
sentation
V, or also for its isomorphism
ling with the quiver
x
V
of dimen-
as a symbol for this repreclass.
For example,
in dea-
£,
the symbol
1
3
1
2
2
1
0
stands for the unique indecomposable
representation
V
of this dimen-
sion type, namely
(111)
+
kkk
kk O/koo ..........> k k k / k o o
k k k / o o k ~ - ' ~ - okk/ook0
].2.
Quivers with relations
We have noted in the last section that for a quiver
gory
MkF
of all representations
category over the path algebra
If
r,s
the form
are two vertices,
of
over
of
F
over
a path from
r
to
(rI~l,~ 2 ..... ~plS)
kF
F
k
~.
i
the cate-
k, defined as follows:
s
of length
where the starting point of
the starting point of any other
F
is just the module
p
~|
is of
is
r,
is equal to the end point of the
146
previous
~i-i' and finally the end point of
Any vertex
r
path algebra
gives
k£
multiplication
and
has as vectorspace
(sI6 l ..... 6qJt),
s.
(rJr).
the product of
The
the
(rle ! .....~pJS)
in this order, will be (ri~ l .....~p,61 ..... ~Jt),
sional algebra if and only if
and contains no oriented
£
Note that
kF
is a finite dimen-
has only a finite number of vertices
cycles
w = (rJ~ 1 ..... ~pJS)
presentation,
is equal to
0, namely
basis the set of all paths,
being the obvious one:
and all other products will be zero.
If
~p
rise to a path of length
is a path in
we can evaluate
w
£, and
on
V
V = (Vi,~)
is a re-
and obtain the linear trans-
formation
w(V)
A relation for
>_ 2
:= ~ p
F
"''''~l
: V
r
---+ V
s"
is a linear combination
of paths of length
with same starting point and same end point, not all coefficients
being zero.
lation
A representation
V of F is said to satisfy the rem
p = ~ K°w.
or to be a representation of
(F,p)
if and only
i=l i l
m
if
m
E Kiwi(V) = O.
i=l
In considering
will assume that all coefficients
speak of a zero relation,
is
I.
relations
K i ~ O.
If
E Kiwi, we always
i=!
m = I, then we will
and we may assume that the only coefficient
Thus, the representations
satisfying a fixed zero relation
are those where the evaluation of a certain path is zero.
A relation of the form
commutativity
relation.
w|-w 2
with
In case
w 2 = (rib I .... ,6qJS), and the
points of the
we call
~i
wl-w 2
Given a set
the category
p+q
MR
Pi
of
(i E I)
£
over
of R-modules,
the (twosided)
be called a
Bi
are pairwise different,
relation.
of relations
k
paths,will
vertices which occur as starting
or as end points of the
a strict commutativity
representations
<PiJi E I>
Wl,W 2
w! = (rJ~ l ..... ~pJS),
for
F, the category of
satisfying
where
p. can be considered as
i
R = kF/<PiJi E I>, with
ideal generated by the relations
Pi"
147
Conversely,
in case
a basic k-algebra
with
rad R
(so that
R/rad R
P
closed field, and
(of course,
R
is of the form
the generators
to us, we
Pi
p
will call the algebras of the form
type.
kF/<O>
r,s
subset of the set of arrows of
s.
Denote by
algebras accor-
In order to simplify the notation,
we will introduce the following convention:
without oriented cycle and
to
of the ideal
is of particular
Our present aim is to classify the one-relation
r
k,
kF/<Pili E I>
algebras.
ding to their representation
from
is
and we may change them con-
The case of one single relation
one-relation
R
is a product of copies of
R), then
are not uniquely determined,
veniently).
interest
is an algebraically
the radical of
for a unique quiver
<Pill E I>
k
Assume
two vertices of
F
F.
is a quiver
Let
FI
be a
F, which contains at least one path
0 = E w.
the relation for
P
which is
I
given by the formal sum of all paths from
r I.
We will see that the representation
with
R = kF/<0>
outside
s
F[.
to
s
along arrows in
type of the category
MR
does not depend on the orientation of the arrows
Thus it is sufficient
(we will draw
r
r
to mark the two vertices
as a black circle, and
and to note only the orientation
s
r
and
as a black square),
of the arrows in
r~.
For example
I
o
c~2 " ~ s 62
stands for eight different
quivers
(obtained by adding all possible
orientations
y,~,e)
together with the relation
to the edges
p = (rI~1,=21s)+(rlBl,B21s) ,
~2~]+626!
or simply and more suggestively
= O, whereas
o
o
cL2V
62
148
stands for four quivers with relation
the first example, the ideal
<p>
ezeiy+B2Biy = O.
of
kF
strict commutativity relation, namely
second example, <p>
tion
Note that in
is also generated by a
a2 ~] = B251, whereas in the
similarly is generated by the commutativity rela-
~2~i Y = 62S]y , but this is not a strict commutativity relation.
Let
F
be a quiver with relations
Pi' i 6 I.
There are several
ways to obtain from these datas other quivers with relation.
quiver
F'
with relations
Pi' j 6 J, is obtained by a sequence of
processes of the following four types, then we will call
a specialisation of
|.
i 6 I.
(F',pj)j
(F,Pi) i.
Adding of relations:
for
If the
let
F = F', and let
Thus, we add additional relations
In this case, kF/<Pili 6 I>
maps onto
PiT
I c_ J, and
pj' with
=
Pi
j E J "- I
kF'/<p~l j 6 J>, thus we have
a full exact embedding
MkF'/<p~Ij E J>
2.
Deleting of vertices:
be obtained from
a.
F
Let
> MkF/<Pili E I>
a
be some vertex of
by deleting the vertex
Also, delete from any relation
les of paths going through
tion
P!'z Let
This defines
J
Pi
the summands which are multip-
be the subset of all
(£',pj)j E J"
V a = o.
F'
a, and call the remaining linear combinai 6 I
with
Pit
The representations of
form the full subcategory of all representations
satisfying
F, and let
a, and all arows containing
o
in
kF'
(F' ,Oj')j E J
of (r'Pi)i E I
V
Thus again, we have a full exact embedding
MkF'/<pjlj E J> - - +
MkF/<Pili E I>
which in this case even gives an extension closed subcategory.
3.
Deleting of arrows.
Let
B
o
~o
a
obtained from
arrow
Pi' and
~.
F
If the path
B = ~q
w = (ri~ ! ..... eplS)
for some
Let
F'
be
Pi"
'
occurs in the relation
q, then we delete this summand from
this way obtaining a relation
of the non-zer~
be an arrow.
b
by using the same set of vertices, but deleting the
PiT
(or zero).
Let
J
There is a full and exact embedding
MkP'/<pjlj C J> - - +
Pi' in
be the index set
MkP/<pili E I>
149
with an extension
are just those
4.
closed image.
(Va,~p)
with
Shrinking of arrows.
The representations
Let
B
o
>o
a
tained from
If the path
for some
F
by deleting
be an arrow.
B, and identifying
(rI~ I .... ,~plS)
F'
be o b -
say
becomes of length
w.1
a
Pi' and
and
b.
~ = ~q
(rI~l,...,~q_|,~q+l,...,~plS).
In this way it may happen that one of the paths,
with a non-zero coefficient,
n
Pi' = K|W] + }~ K.W. where the
i=2 i i
Let
the vertices
occurs in the relation
p
w!
(F' 'Pj')j E J
b
q, then replace this path by
K i ~ o, and
of
~p~ = o.
w, occuring
in
1, say
are pairwise different paths, all
is of length
I, given by the arrow y. Then we deN
lete y in F' and have to replace ¥ by - I KTIKiwil in any path
i=2
occuring in any relation
p!.
In this case, we obtain a surjection
J
kF/<PiJi C I> .....
C J> and this gives rise to a full ex-
kr'/<p~Jj
act embedding
Mkr'l<p~lj
E J>
~ MkF/<PiJi E I>
which again has as image an extension closed subcategory.
the representations
V = (Va,~)
of
for which
(F''P~)i C J
~
Note that
are just those representations
is an identity map.
The first three processes of adding relations
or deleting points
or arrows are rather familiar to anyone, and usually very easy to
detect.
So let us give just an example for the process of shrinking
arrows which is of interest for our study of one-relation
algebras:
Consider the quiver
and recall our convention
relation
~3~2~I+B352BI
we obtain the quiver
that this means that we are working with the
= o.
If we shrink both the arrows
~3
and
B3'
150
T
C~2 ~
(with relation
and
BI
~2 ~] = B251).
B2
A further shrinking of the arrows
~!
leads to a quiver
o
of type
D"4.
(Here, we first have obtained
T
o
A
~2 ~
~2 = B2' so that we can delete
o
with
B2
together with the relation).
B2
There is an additional process, namely the dualisation.
case, we reverse all arrows of
F
corresponding linear combination
kF'/<0~[i E I>
the category
1.3.
and replace any relation
p'
of the reversed paths.
is just the opposite algebra of
MkF'/<P![ii E I>
In this
~
by the
Then
kF/<0ili e I>, and
is the dual category to MkF/<Pili E I>"
Splitting zero relations
We want to show that certain zero relations may easily be removed.
This process is rather well-known, we will follow the presentation
given by Zavadsky
Assume
F
[39].
is of the form
2
(this should mean that
F' = ~
F
is the union of some subquiver
which is not specified, and the subquiver
161
o÷o÷~ ... o~o÷o with the orientation of one edge not specified, and
o
that these two subquivers intersect in precisely two vertices named 0
and
n+l, but in no arrows).
p : ~n...~l
Define
We are interested in the relation
= O.
A
to be the quiver of the form
(with the same
:
4"
~
o (n-l)'n'
3"i,, o~
F', and an orientation of the edges
responding to that of
representation
V = (Vi,~e)
and
B'
and
B"
cor-
B).
Consider the canonical functor
WIp, = VIF,
2''
~ : MkA + Mkr
of
A
which associates to a
the representation
W i = Vi, • Vi,, for
] < i < n
W
with
(and correspondingly
L
i
forming direct sums of the given linear transformations).
Lemma
V C ~A
I:
A representation of
kF
is of the form
if and only if it satisfies the relation
p.
~(V)
for some
The functor
induces a bijection between the indecomposable representations
MkA
and in
Vn+ 1 ~ O.
Wo = 0
MkF/<p>
For any indecomposable representation
and
sentations
which satisfy (in either category)
in
MkF/<p>
V
in
or
with
Wn+ 1 = O, there are precisely two indecomposable repreV
in
~A
with
~(V) = W.
Thus it follows that the functor
pairs
W
V° # O
of indecomposable representations
number o f i n d e c o m p o s a b l e r e p r e s e n t a t i o n s
1
3
4
0"----'~0
0
2
~
identifies precisely
in
Mkk, s i n c e
of
n-1 n
o---~o
n(n-1)
n(n-l)
is the
152
Proof:
striction
Let
W
be indecomposable
to the subquiver
1
3
4
c~ 2
in
n-1
n
its re-
n+l
c~ 3
0
Consider
MkF/<p>.
F"
~,:~
c~n
~0
• . .
0
)0
)0
f
o
2
of type
Dn+ I .
composable
Xn+ 1 = O.
Let
WIF, = X @ Y, where
F"-representations
Case by case inspection
tions with non-zero
of the maps
is a direct
sum of inde-
(n+l)-component,
of the indecomposable
(n+l)-component
~n...~2
Y
with non-zero
shows
is a monomorphism.
that in
Thus,
Y
and
F"-representathe composition
since
~n...~l
= O, we
see that
Wo
maps
to
into
F'
X I.
being
Let
V
WIF,,
~ W1 = X1 @ Y1
be the representation
to the left arm being
Y, and using the maps
~I
XI
: Wo
~(V) = W, and this is the unique
or
W ° # O.
If both
and
representation
V
of
restriction
V
now to the right arm.
of
Similarly,
let
\,,./
/<C_//
F
with
with restrictions:
X, to the right arm being
~ n : Yn ÷ Wn+l"
be of the form
C~l /
°'2
... oC~n- 1,o
2
o
B1
1
o
p : ~n...~l
= O.
c~n
:$£.;i;:f
Then
in case
Y = O, and there
~(V) = W, namely with
m~m
again with relation
A
such representation
W ° = 0 = W + I i then
A
of
X
Wn+ 1 # O
is a second
being the
153
Define
A
o
I'
to be of the form
o
...
2'
and the functor
O - ..........
~
(m+l)
m"
I (m+2)'
(m+n-1)
' *
(m+n-2)" I
i (m+n-2) '
(m+2)" i
o (m+n-l)'
(m+l)" 6
Mkr/<~>
~ : MkA ÷
~(V) i = Vi, 0 Vi,, for
with
o ... o
m"
2"
o
l"
and
~(V) Ip, = VIF,
I J i j n+m-l.
Again we have
Lemma
2:
The functor
~
induces a bijcetion between the indecompo-
sable representations
V
V° ~ O
For any indecomposable representation
or
MkF/<o>
Vn+m ~ O.
with
in
MkA
and in
MkF/<p>
which satisfy
W
in
Wo = O = Wn+m, there are precisely two indecomposable
representations
V
in
MkA
Thus, here the functor
with
~
~(V) = W.
identifies precisely
pairs of indecomposable representations in
l
~(n+m)(n+m-l)
MkA, and leaves the re-
maining ones distinct.
Proof:
In this case, the restriction of a representation
to the subquiver
W
of
F
r"
m+l
m+2
I
>o
...
n+m-1
n+m
o
)o
~2
an
mo
1 o
decomposes
phism in
Xm+ I.
Wit, = X ~ Y
Y.
Then again
where
Xn+ m = O
and
~n...a2
~I : We ÷ Wm+l = Xm+l @ Ym+l
Define the representation
V
of
A
as follows:
is an
monomor
maps into
VIF, = W[F,,
154
and the restriction
being
to the left arm of
being
We call relations
all arows reversed)
of the two types above,
splitting
zero relations.
and the dual ones
In considering
(with
the
type of a quiver with relation we always may assume
that no splitting
zero relation occurs.
the only ones which can be separated
Note that these relations
in such an easy manner.
in all other cases of a single zero-relation
exists an indecomposable
with both
X, to the right arm
~1 : W o ÷ Xm+1' a n : Y n+m-l ÷ Wn+m"
Y, and use
representation
A
Vr # 0
and
representation
V s + O.
Clearly,
V
are
Namely,
(rl~I,...,~plS) , there
satisfying
this relation
we only have to consider
the
following cases:
o
and
Of course, we can choose an arbitrary
Examples
of representations
V
in
orientation
MkF/<p>
of the free arms.
with
Vr ~ 0
and
Vs + 0
are as follows:
ok
(II)
. . . . . . .
~///y
/oo~
{110)
(011)
J
/~OK
//
155
~FM~//,,?//,,A////;,~/.///:///7//,,/
m,
]
....
........
,,..t2/
(010I)
(lOll)
1.4.
The representation types
Of course, a finite dimensional algebra
finite representation type provided
MR
R
is said to be of
is finite (has only finitely
many indecomposable objects).
We will say that
that
R
is wild)
R
is of wild representation type
(or just
provided there is an exact embedding of the cate-
gory of representations of the quiver
into
MR
which is a representation equivalence with the corresponding
full subcategory of
to be full).
MR .
(Note that we do not assume the embedding
The path algebra
k~
of
~
is just the free associa-
tive k-algebra with two generators, also denoted Sy
k<Xi,X2 >.
The
reason for calling such algebras wild stems from the fact that for
any other finite dimensional k-algebra
bedding
Mk~
R', there is a full exact em-
M R , + Mk~ , in particular, there are full exact embeddings
÷ Mk~
where
~n
is the
n
arrow quiver
n
for any
type, see
n.
For a discussion of categories of wild representation
[ 6, 19].
Finally, the algebra
type
provided
R
R
is called to be of tame
representation
is not of finite representation type, whereas for
156
any dimension
Fi : ~ [ T ]
R-modules
d, there is a finite number of embedding functors
÷ MR
such that all but a finite number of indecomposable
of dimension
some indecomposable
almost all
Fi(M)
d
are of the
k[T]-modnle
M.
are indecomposable
form
Fi(M),
(independently
of
i, and
Fi,
and pairwise non-isomorphic,
then we will call this set a series of R-modules.
exists
for some
Note that if for some
In case there
d) a finite number of such embedding functors
F. such that, for any dimension
d, all but a finite number of indei
composable R-modules of dimension
d are of the form Fi(M) , then R
will be called domestic.
Of particular
interest will be embedding functors
F i : Mk[T] --+M R
which are in addition full. As we will see, for the
tame one-relation
algebras
studied in these notes, always
will exist. In this case, the irreducible
under
F. to a one-parameter
l
endomorphism ring k.
k[T]-modules
family of indecomposable
such functors
are mapped
R-modules with
Examples of domestic algebras are the path algebras of tame quivers. In case one deals with a connected
tame quiver
delete the images of one full embedding
functor
F : J~k[T] --+ Mk£
order to remain just a finite set of isomorphism
sion. Also,
have been considered
in [29, 33, ]3].
tame algebras,
k[TI,T2 ]./<TI,T2_a
_b>
a ~ 2, b ~ 3, studied by Gelfand and Ponomarev
in
classes in any dimen-
there are known examples of non-domestic
first one seems to have been the algebras
F, one only has to
the
with
[21]. Further examples
157
1.5. The classification of the tame one-relation algebras
Theorem
les, and
If
p
I: Let
£ be a connected quiver without oriented cyc-
a relation for
~ which is not a splitting zero relation.
(F,p) is tame, then it is a specialization of one of the following
tame one-relation algebras or their duals.
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1
We have first listed the quivers with a zero relation,
with those without cycles,
followed by those with cycles.
starting
Next, there
are listed those algebras which are given by one commuting cycle with
additional
arms, and then the remaining quivers with a strict commuta-
tivity relation.
After this, there are the quivers with a commutati-
vity relation which is not strict,
involving
three different paths.
and finally quivers with a relation
For all types, we have grouped to-
gether the algebras with similar categories of modules.
167
Always, we have noted the representation
relation algebras.
rather detailed
type of the given one-
In fact, for the domestic algebras, we will provide
information on the whole module category,
tinguish here four different possibilities.
refers to a concealed quiver of type
~
First,
and we dis-
the symbol
in the sense of
the first ten algebras all are concealed quivers).
C~
2.3
(thus,
We will see that
the module category of a concealed quiver is rather similar to that of
the corresponding
quivers,
R~
quiver.
All other algebras are enlargements
and we denote by
F~
a finite enlargement
a domestic regular enlargement
(see
there exists precisely one one-parameter
vial endomorphism
ring.
3.7).
(see
In these cases
The last domestic
case to be considered
there are precisely
two one-parameter
endomorphism
This case will be denoted by
ring.
will be considered
The non-domestic
in
~
G~-~,
referring
algebras are denoted by
regular enlargements
type
~
N.
to
and
There are two types of
separately
in
3.9.
as considered
in writing
linear combinations)
on which a corresponding
quadratic
All others
in part
3, and
N~.
Also, we have listed positive vectors which generate
integral
~
2.7, these are the domestic cases.
we have added the similarity
negative
is
so that
families of modules with trivial
with a quiver of type
algebras which have to be considered
are non-domestic
C,F,R,
family of modules with tri-
the case where two tame connected quivers are glued together,
the glueing of a quiver of type
of tame
2.6), by
(using non-
the set of all positive vectors
form takes value zero
([ 7]).
Note that in case the algebra is obtained by glueing two tame quivers
together
(case
G),
of the quadratic
only the multiples
form, whereas
for the non-domestic
dratic form is positiv semi-definite,
form are closed under addition.
of the given vectors are roots
algebras the qua-
so that the roots of the quadric
It is rather easy to see that all
listed vectors actually are dimension vectors of modules with trivial
endomorphism
ring belonging
Theorem
les, and
O
2:
Let
£
be a connected quiver without oriented cyc-
a relation for
representation
to a series of such modules.
£°
If
(F,p)
type, then it specializes
splitting zero relation,
algebras or their duals.
is not of finite or tame
either to a wild quiver with
or to one of the following wild one-relation
168
Quivers without relation:
D4
ZII
S.TIII
~6
ZIII
S.TIV
~7
ZIV
~8
zv
&TVI
S.TV
Quivers with one relation:
I
ZVI
II
ZVII
III
c~
ZVIII
IV
Q
ZIX
V
c~'7
ZX
VI
C~7
ZXI
VII
C~'7
ZXII
c~7
ZXIII
VIII
o~~
169
IX
ZXIV
X
CL~7
ZXV
XI
C~8
ZXVI
c~ 8
ZXVII
Xlll
c~ 8
ZXVIII
XIV
CE 8
ZXIX
XV
G
ZXX
XII
OD-~O~'O
XVI
o-o-~.~
ZXXI
XVII
XVIII
0~0--0~0--C
XIX
0"~--0~0~
?
XX
~o
XXI
7~II
~ 0--0--0--0
c~ 8
ZXXII
CE 8
ZXXIII
-o
ZXXIV
c~ 8
ZXXV
c%
ZXXVI
CE 8
ZXXVII
?
170
XXIII
~12
XXIV
~13
XXV
XXVI
XXVII
XXVIII
c~7
S.TX
c~8
S.TVII
XXX
c2 8
S.TIX
XXXI
c2 s
S.TVIII
XXXII
c~8
S.TXII
XXXIII
c2 8
S.TXI
7~XIV
c2 8
S.TXIII
XXIX
0~--0~0~0-~0--0
171
~I
XXXVII
~2
F =
c~2~I-B2BI
Bl
PAl l
B2
Four of the wild quivers are denoted by
~4' ~6' ~7' ~8' since
obviously they relate to the corresponding extended Dynkin diagram in
the same way, as these extended Dynkin diagrams relate to the Dynkin
diagrams
D4, E6, E7, E 8.
Most of the wild quivers with relations
listed above are concealed quivers in the sense of
C.
2.3, the symbol
indicates that we deal with a concealment of a quiver of type
Some algebras are regular enlargements of a quiver
they are denoted by
R., with
.
F
(see part
denoting the type of
F.
..
3),
There are
four cases which are neither concealed quivers nor regular enlargements of quivers, and which have to be considered separately.
Theorem
I
and
2
essentially are due to Shkabara and Zavadskij
who considered the following two important special cases:
[40]
Zavadskij
classified the quivers without cycles (oriented or not) with a
single relation (which therefore has to be a zero relation) which are
tame.
In our lists above, we have added the corresponding numers of
Zavadskij's
list with the symbol
Z.
(Since Zavadskij's list con-
tains with any tame algebra also its specialisations, not all numbers
from his list appear here).
Similarly, Shkabara
[36]
has classi-
fied the quivers with one strict commutativity relation which are tame,
we refer to his list by the symbol
S.
Note however, that Shkabara
does not exclude oriented cycles, so his result is more general, but
in essence, he does not obtain additional algebras.
[24]
Also, Marmaridis
has considered the algebras which are obtained from a commuting
cycle by adding arms, and he determined the representation type in all
but three cases
(54,55,56).
172
;.6.
Outline of ~roof
The proof of theorems
!
and
2
uses three different types of
arguments.
(a)
theorem
We have to show that the one-relation algebras listed in
1
are tame.
This will be the most interesting part, and we
will use most of sections
2
and
3
for this part of the proof.
Besides developping some rather general methods we will try to give a
good insight into the actual behaviour of the corresponding module categories.
In particular, we will explain the different ways of beha-
viour marked in the list by the letters
(b)
C,F,G,
...
We have to show that any one relation algebra which is not
a specialisation of one of those listed in theorem
sation of one of the forms listed in theorem
2.
natorial part of the proof, and rather technical.
ments will be given in the next paragraph,
similar
(but perhaps even more boring):
|
has a speciali-
This is the combiSome of the argu-
the remaining ones are very
also, they may be found in the
literature.
(c)
2
Finally, one has to show that the algebras listed in theorem
are wild.
This is the easiest part of the investigation,
so let us
use the remainder of this section to write down some of the embeddings
~
÷ MR
which show that the corresponding algebras
For the quivers without relations,
[12,26]:
R
are wild.
these embeddings are w e l l - k n o ~
For example, given a representation
(V,~,@)
of
~, define
representations
id
V
of
o ~ o
~0
,
and
(0)
~ V
o÷O,
and
respectively.
note the direct sum of two vectorspaces
given a representation
U --+ V ~ W
following representations
V
V
of
and
id
> VV
VV
Note that we will deW
o÷~p,
just by
VW.
Then,
we consider the
173
1
T
VO +--
UO
r(~)
or
or(v)
F(~)O
where
F(~)
--+ VWO
--+ V~V *-- VOV ~--
denotes the graph of
{(vov) IvCV} +- {(uou) luC~},
~ : V + W, as a subset of
whatever is more convenient, and similarly for
~7
example the embeddings of the representations of
presentations of quivers of tpyes
of
~7
and
and
~
VW
or
WV,
~8' using for
into the re-
~8' as listed in the tables
[11].
The algebras
I - XXII, and
XXVIII - XXXIV
are concealed qui-
vers, with obvious concealments of quivers of types
~8' see
2.3.
~4' ~6' ~7' and
For the remaining cases, we have to define again indivi-
dual embeddings.
The algebras
lar.
Note that
XXIII, XXIV
XXXV
and
XXXV
can be defined also by a zero relation, namely
in all three cases, we define functors from
a quiver of type
can be treated rather simi-
MkF
into
MR, where
F
a cofinite subcategory of
MkF
(defined by the requirement that the
,maps are either monomorphisms or epimorphisms, as indicated) and the
image category in
is
~8' which will be a representation equivalence between
MR .
Namely,
174
U
o
l
UI---+U2---+V--+~U3+--
U4÷--
U5---++U6+--
U7+--
U8
shall go to
woo
(o*o)
U] --+ U2-----+VU5U 8
similarly,
~UoU4U7~U3U6,
~n the case of
XXIV,
~o
U I + ~ - U2~+-- V
>> U 3 +--- U 4 <
U5+--
U6
'> U 7 +--- U 8
shall go to
~o
(o*)
U
o --+ VU 6
and finally
......~. U2U 5
in the case of
U
U1
shall go to
~+ U 2
÷ V
~ U]U4U 8
> U3U 7 ,
XXXV,
o
->+U3 +
U4
-++U5 ÷
U6
-~U 7 +
U8
175
,/*oo~,
0~0
t,oo,/
(;)
u1--+ u 2 ~.~
.7
V
~ U3U5U7
U4U6U 8
U
o
In the cases
XXV
and
o--+~-~,~, namely we send
XXVI, we define again a functor from
qO
U ÷V ~
W
to
$
qo
V
oVW
> W
7\
r(,)
ow
!
r(,Iu)
and
> V
~VW
/\
r (,)
In the cases
~W
XXVII
and
ow
XXXVI, we define a functor from the
~4-quiver with subspace-orientation:
we send
u2
U1
> V<
u3
U4+---
U5
176
to the representations
id
o*)
(,o)
U1U5
U2U 3
and to
(o,)
V/U I @ U 5
> U4
V
v
id
Note that in the last case, we even have obtained a representation of
~
'° ~ 2 ~
with
~3"2~i = B2BI = O-
It remains to consider the case
directly
an embedding o f
Mk~
into
XXXVII.
MR .
onto
(~)
V
id
W
(~)
W
(ol)
This time, we define
Namely, we s e n d
.
(V,~,~)
177
1.7.
Let
The combinatorial part of the proof
(F,p)
be a quiver with one relation
splitting zero condition.
p
We assume that neither
which is not a
(F,p)
nor its dual
has a special!sat!on to one of the quivers with or without relation
given in theorem
2.
We have to show that then
(F,p)
is a special!-
sat!on of one of the quivers with relations given in theorem
Let
lation.
r
be the starting point, and
Since the quivers
F~{r}
s
and
1.
the endpoint of the re-
F'-{s} both are specialisations
of
(F,p), they have to be disjoint unions of quivers of types
m
Let p = E K.w. with pairi=! ~ ~
wise different pathes wi, and all K i # o.
A n , Dn, E6, E7, E8, An, Dn, E6, E7, E 8.
Consider first the case where two of paths
w.
have an arrow
i
in common:
~i = Bj
say
w I = (rJ~ I ..... ~pJS),
for some
i,j.
Since
w 2 = (rI~ I ..... BqJS),
and
w] # w2, we can assume
(~i+] ..... ~p) ~ (Bj+] ..... Bq), otherwise we consider the dual situatiom
Now
of
F~{r}
has a subquiver of type
F~{r}, and therefore
possible paths
~n' thus this must be a component
i = j = ].
with starting point
It follows, that
(F,p)
Also, w]
r
and
w2
and endpoint
are the only
s, thus
m = 2.
is of the form
c~3
Ir
Note that
F'
B2~
>o
o ~
s "
has to be a tree, since otherwise we obtain a specia-
l!sat!on of the form
~p-+o.
If
F'
contains a subquiver of the
form
)O
• • •
O
>O
,
r
then
(F,p)
fore see that
specialises to
(F,p)
XXX.
In case
is a special!sat!on of
p = 2
74.
or
q = 2, we there-
However, if both
178
p > 2, q > 2, then we use that the connected
embedded
into one of
Dn, E6, E 7
ties are specialisations
of
or
subquiver
F'-{s} can be
~8' and then the only possibili-
75-82.
Thus, we can assume that no two paths
mon.
Since there is no specialisation
w. have an arrow in comi
of the form XXXVII, we see
that they also cannot have any vertex but starting and end point in
common.
Consider now the case
m > 3.
If we shrink all arrows occuring
in
wI
but one, then we can
delete the remaining one together with the relation.
quiver
F'
without a relation containing
~pq, thus this subquiver must be all of
(F,p)
Since
a subquiver of the form
F', and therefore
m = 3, and
is of the form
r
that
We obtain a
F~{r}
(F,p)
~ o
+o
s
must be embeddable
into
is a specialisation
In case
commutativity
m = 2, the relations
relation,
starting point of
say
~i' and
Dn,
p
P = ~p'''~l
bi
E6, E 7
of one of
or
~8' we see
83-86.
can be assumed to be a strict
- Bq'''BI"
Let
the starting point of
ai
be the
Bi, thus we have
a subquiver
a2
~2
a3
alp
r=al=b 1
o
s
(,)
bq
If there is any additional
hi, then we obain
additional
O
(non-oriented)
(non-oriented)
to
path joining the
as a specialisation.
path joining one of the
a.
I
ai, or the
If there is an
with one of the
179
b.
(with i,j ~ 2), then we obtain XXXVII as specialisation. Finally,
J
if there is an additional (non-oriented) path joining
r with
s,
then
XXXVI
shows that
since any additional
cialisation.
(~)
p=q=2,
and then we clearly deal with case 75~
arrow would give
~
or the dual as spe-
This shows that we can assume that
F
by only adding arms at the various vertices.
vestigation
ridis
of this situation we refer to
is obtained
from
For a detailed in-
Shkabara
[36]
and Marma-
[24].
Similarly,
of the form
Thus,
tion and
in case
m=1
and
does not contain any subquiver
of Zavadskij
it remains to deal with the case that
F
contains a (non-oriented)
p = (rI~ ! .... ,~plS).
7
c I c2
F.
p
is a zero rela-
Thus, let
cj +2
cn
be a cycle in
cycle.
[40].
Also let
Cn_ |
Since
£
Apq, we refer to the investigation
c j+|
cj
First, assume that
r
does not belong to the cycle.
F'~{r} contains a cycle, this cycle must be a component of
F~{r}, thus
r
is the neighbor of one of the
c..
Therefore,
we may
i
assume
o----+o . Also, ~ = ~I' since otherwise
the deletion of
r
c!
gives a quiver without relation which has a specialisation of the
form
o--+q~-~.
Thus
(P,p)
is o~ the form
! c2
cj=s
Since there is no specialisation
that
j J 3, and
F' + {r}
then we deal with the case
£' is a subquiver of ~
tion of
s
XXIII
or
If
29.
XXXV
shows that for
Finally,
.°. O-~r, thus
(F,p)
j=3
XXIV, we see
j=2.
and
r'={r},
j=2,
is a specialisa-
31.
Next, assume that
also
of the form
implies even
r
belongs
belongs to the cycle.
to the cycle, and, by duality,
Note that the path
w
has to be part
180
of the cycle since otherwise we will obtain
zation.
splitting
We want to show that
zero relation,
specializes
p=2.
~
as a special
Since the relation is not a
the assumption
p ~ 3
shows that
(F,p)
to one of the following quivers with relations:
a)
b)
In case
a), we obtain as further specialization
which is a splitting
Case
b)
which again
Case
c)
case
d)
the case
to
is a splitting
ist
just
XXVII
the deletion
of
zero relation
of wild
and therefore
impossible,
a3
gives
a quiver
type,
to type
namely
whereas
~4"
~8 o
in
Now i n
p=2, we cannot have
asspecialization
specializations
see that
zero relation of wild type.
specializes
(£,p)
(since this further specializes
of the forms
XXV, and
has to be of the form
~XVI
31.
to
~4 ).
Also,
being impossible,
we
181
CONCEALMENTS,
2.
FINITE ENLARGEMENTS
2.1
The Auslander-Reiten
Let
XR' YR
f : X÷Y
quiver
be modules over some ring
is called irreducible
in case
morphismnor a split epimorphism,
f = f2.fl, we have that
epimorphism.
non-trivial
AND GLUEING
f|
f
R.
A homomorphism
is neither a split mono-
and if for any factorization
is a split monomorphism
or
f2
is a split
Thus the irreducible maps are those maps which have no
factorizations.
In order to be able to speak of the mul-
tiplicity of irreducible maps between two modules of ~nite
have to introduce the abelian group
length, we
irr(X,Y).
n
mLet
X, Y
be R-modules
of finite length,
say
X = J@IXi'
Y =
@ Y., with X. Y. indecomposable.
A homomorphism
f : X ÷ Y
i=] i
l'
i
is said to belong to the radical
rad(X,Y)
provided no component
fij
: Xj ÷ Yi
eription of
with
is an isomorphism,
f.
Let
f = f2.fl}.
is irreducible
rad2(X'Y)
For
X,Y
f = (fij)
indecomposable,
if and only if
situation we call
where
is the matrix des-
= {f13 fl E rad(X,Z),
f2 C rad(Z,Y)
a homomorphism
f C rad(X,Y) ~ rad2(X,Y),
irr(X,Y) = rad(X,Y)/rad2(X,Y)
f : X ÷ Y
and, in this
the group of irredu-
cible maps.
in case
closed field
space over
R
X
to
Y.
the isomorphism
number of arrows from
[X]
X,Y
the multiplicity
The Auslander-Reiten
[X]
to
[Y]
irr(X,Y),
R
has as
R-modules,
is precisely
modules,
vector-
of irredu-
quiver of
class of the module
of indecomposable
tives of a basis of
algebra over an algebraically
is a finite dimensional
classes of indecomposable
denotes the isomorphism
every pair
irr(X,Y)
k, and we call its dimension
cible maps from
vertices
is a finite dimensional
k, the group
and the
dim irr(X,Y), where
X.
If we choose for
a fixed set of representa-
then we call these maps fixed irredu-
cible maps.
One can obtain a rather large amount of information
the Auslander-Reiten
Assume that
R
quiver from the Auslander-Reiten
is a finite dimensional
at least, an artin algebra).
algebra over some field
(or,
Recall that an exact sequence
f
o --7 X---+ Y
is an Auslander-Reiten
concerning
sequences.
Z ---+ o
sequence if and only if both
(*)
f
and
g
are
182
irreducible maps.
universal
sequence
properties:
The sequence
the following
to
Y:
Auslander
B : Z' ÷ Z
module
say
Z
which
sequence
(*)
and
are isomorphic
(*)
and
(**)
~ = ~'-f.
is not a split epimorphism
B' : Z' ÷ Y
with
module
~ = g-~'.
to isomorphism,
and we call
Z, and also write
and unicity
which is not
(*)~
for any
there exists an
g'
~ Y'----+ Z'---+ o
iff
and
sequence
Z = A-(X).
namely
Z
X = A(Z)
in certain quotient
cit construction,
X
sequence
Z'
are equivalent.
two ends of an Auslander-Reiten
functorial
which
with
se-
and
X'
the sequences
(*)
If we have two A u s l a n d e r - R e i t e n
f'
of
: Y + X'
is not projective,
(*).
and
is not a split monomorphism,
~'
For any indecomposable
o --+ X'
X
which
there exists
and Reiten have shown both the existence
Auslander-Reiten
then
is an Auslander-Reiten
are indecomposable
there exists an Auslander-Reiten
indecomposable
quences,
Z
conditions:
there exists
of these sequences:
injective,
(*)
and
~ : X + X'
Y:
Any h o m o m o r p h i s m
can be lifted to
X
equivalent
Any h o m o m o r p h i s m
can be extended
(ii)
they have been defined by the following
if and only if both
satisfies
(i)
Originally
are isomorphic
In particular,
(*)
determine
Note that
A- = ~
A
and
A-
iff
the
each other up
the Auslander-Reiten
categories,
A = D Tr,
(**)
translate
are only
but that there is an expliD, where
Tr Z
denotes
the
"transpose"
of Z
(form a minimal projective resolution
h
Q - + P --+ Z --+ 0 of Z, and let ~ Z be the cokernel of
Hom(h,R)
: Hom(P,R)
d u a l i t y with respect
Reiten sequences
Lemma
|:
--+ Hom(Q,R)
,
whereas
to the base field.
and irreducible maps
Let
field, and let
R
D
denotes
the ordinary
The relation between Auslander-
is given in the following
be an algebra over an algebraically
closed
Ill!
ft
0 ---+ x
Y @ Y' \i=l "
lemma:
> Z ----+ O
183
be an Auslander-Reiten
sequence, with
Y
has no direct summand isomorphic
to
~] ..... ft 6 rad(X,Y)
= irr(X,Y)
is a basis of
/ rad2(X,Y)
quence starting with
lander-Reiten
completely
X
X
is a basis, and
non-injective,
determines
the Auslander-Reiten
with
[X]
in case
[Z]
L emma
I
X
Let
Also, for
Z
Z
non-
determines
quiver with end point
even easier way to obtain all arrows starting
Z
R
se-
the arrows in the Aus-
[X].
sequence ending with
is indecomposable
in case
2:
Let
g] ..... gt
the Auslander-Reiten
completely
the arrows in the Auslander-Reiten
There is a similar,
field.
Y'
Then
quiver with starting point
[Z].
ending in
such that
irr(Y,Z).
This shows that for
projective,
Y.
indecomposable,
injective,
is indecomposable
and all arrows
projective,
as follows:
be an algebra over an algebraically
be indecomposable
closed
injective with socle socI.
Let
li
ft
f,
0 ---+ soc I
--+ I
i=!
be exact, with
Y
indecomposable,
mand isomorphic to
Let
P
Y.
Then
such that
fl,...,ft
be indecomposable
Y'
has no direct sum-
is a basis of
projective with radical
irr(I,Y).
rad P.
Let
(gl,---,gt,g')
@ Y~ • Y' -, P--+ P/mad P --+ 0
~i=l i
0---+ (
be exact, with
Y
mand isomorphic to
indecomposable,
Y.
Then
such that
gl ..... gt
Y'
has no direct sum-
is a basis of
As a consequence we see that the Auslander-Reiten
finite dimensional
algebra
ways is locally finite
R
quiver of a
closed field al-
(any vertex is starting point or end point of
at most finitely many arrows),
or countable.
over an algebraically
irr(Y,P).
In fact, in case
thus its connected components
R
are finite
is connected and not of finite
184
representation
type, then no component
Another consequence
composable,
[X]
is finite
of the previous assertions
is a source if and only if
and a sink if and only if
X
(see [2],or also [35]).
is that for
X
X
inde-
is simple projective,
is simple injective.
Gabriel and Riedtmann have proposed
tion to the Auslander-Reiten
to consider in addi-
quiver also a two-dimensional
cell com-
plex, derived from the underlying graph and the action of the Auslander-Reiten
translate
A, at least when all multiplicities
irreducible maps are
Auslander-Reiten
responding
the
Namely,
the points are the vertices of the
there are two kinds of edges:
to the arrows of the Auslander-Reiten
orientation),
non-projective,
Finally,
< I.
quiver,
and, in addition,
the ones cor-
quiver
(just forget
for any point [Z], with
there is an additional
there are triangles
of the
edge between
AZ
and
Z
Z.
of the form
[AZ]
[z]
in case we have the Auslander-Reiten
sequence
S
0
with all
Yi
> AZ ----+ @ Y. ---+ Z
i= 1 1
indecomposable,
~ O,
the boundary edges of such a triangle
being the edges corresponding
to the arrows
[AZ] + [Y.]
and
i
[Yi ] ÷ [Z]
between
in the Auslander-Reiten
[AZ]
and
Auslander-Reiten
[Z].
quiver, and the additional
edge
Note that the connected component of the
quiver give rise to the (topological)
components
of
this cell complex.
In case some multiplicites
vided the Auslander-Reiten
0-----+
X
are
sequences
>
Ill
.
s
s
. . ÷
@ Y.
i=!
I
I, we proceed similarly,
pro-
in some component are of the form
(gl ..... gs )
~ Z ----+ 0
185
with
Yi
indecomposable and fixed irreducible maps
fi,g i.
Here we
construct a cell complex only for this particular component.
there will be
s
Again,
different triangles for the Auslander-Reiten se-
quence above, which, as before, have the one boundary edge between
[X]
and
[Z]
the points
some
in common, but which in addition may also have some of
[Yi ]
in common, namely in case
Yi,Yj
are isomorphic for
i ~ j.
2.2
Quivers without oriented cycles
The representation theory of quivers without oriented cycles will
play a dominant role in the further investigations.
Here, we present
the structure of the components of the Auslander-Reiten quiver which
contain projective or injective modules.
quiver without oriented cycles, and
If
[a,b] = {z E ~la < z < b}
a E ~ U {-co}, b E Z U {~}, and
following quiver:
io< ~
Thus, let
be a (finite)
is some intervall in
a < b, we denote by
[a,b]F
Z, where
the
F × [a,b], and for any arrow
(i,z) (~,z) ~(j,z)
(j,z) (~*'z))(i,z+1), for
P
some commutative field.
its set of vertices is
~ , there are arrows
and arrows
k
for any
a < z < b.
a j z J b,
Let us give some
examples:
<<
I = [1,6 ]
\
~o
I=lq
I=l~-
where
~
"'"
. . . ~
~ = {1,2,3,...}
are the natural numbers, and ~ - = {-zIz E ~ } .
186
Assume now that
F
is a quiver of type
We are going to define a subquiver
A(F)
of
An, Dn, E 6, E 7, or
~F
E8.
as follows. Let
note the following permutation of the vertices of
F:
be the identity. For
F
of type
n ~ 0(2), or of type
An, with
n
E 7 or E 8, let
arbitrary, Dn, with
o
n m ](2), or
F
de-
Dn,
with
If
a
is of type
E6, let
be the unique non-trivial automorphism of the underlying graph of
obtains the underlying graph of
For any vertex
let
i
of
F
F
(one
by replacing the arrows by edges).
F, there is an unoriented path from
i
to
o(i),
bi
a. be the number of arrows in this path directed towards i, and
1
the number of arrows in this path directed towards o(i). Also, denote
by
h
for
the "Coxeter number" for
£
of type
~F
of all vertices
z ~ ½(h+ai-bi). We give some examples:
r
J
h = n+l, 2(n-l), 12, 18, or 30,
A n , D n, E 6, E 7, or E 8, respectively. Define now
as the full subquiver of
]
F, thus
Aft)
(i,z)
satisfying
A(F)
187
Lemma.
For a quiver of type
Auslander-Reiten quiver of
kr
An, Dn, E6, E7, or E 8, the
is of the form
A(F). If
F
is a
connected quiver without oriented cycles, and not of the form
An, D n,
E6, E7, or ES, then one component of the Auslander-Reiten quiver of
contains all indecomposable projective modules and is of the form
kF
INF
(the modules in this component are called preprojective), another component of the Auslander-Reiten quiver contains all indeeomposable injective modules and is of the form
~F
(the modules in this
component will be called preinjective).
Reiten translate is
In all cases, the Auslander-
(i,z) + (i,z-]).
Thus, the modules corresponding to the vertices labelled
in the Auslander-Reiten quiver of a quiver of finite type
An, Dn, E6, E 7
and
(r,])
(the cases
ES) , and in the preprojective component of the
Auslander-Reiten quiver of a connected quiver of any other type, are
just the projective modules.
In fact, we can describe this correspon-
dence in more detail as follows:
P(r)
For any vertex
the following representation of
space with basis the set of paths
and for any arrow
by
~---+~ in
F:
F, let
~
of
P(r) s
(rJ~i,...,~pjS)
F, denote by
be the vector-
from
: P(r) s --+ P(r) t
(rI~ ! ..... ~pjS) ÷ (rI~ ] ..... ~p,~jt).
P(r) = (P(r)s,~ ~)
Let
r
r
to
s,
be defined
This representation
is indecomposable projective, and it corresponds to
the point
(r,]). Clearly, if ~-~+~O
is an arrow in r, then P(j)
i
j
is isomorphic to a direct summand of the radical Tad P(i) of P(i),
an embedding
(ilBIj).
~B : P(J) ÷ P(i)
In this way,
being given by left multiplication with
rad e(i) =
@ ~Bp(j). But this shows that
B :i~j
the full subquiver of the Auslander-Reiten quiver generated by the indecomposable projective modules is just the quiver
obtained from
F
[],]]F, which is
by reserving the orientation of all edges.
In the
same way, the full subquiver generated by the indecomposable injective
modules is also of this form.
From the knowledge of the indecomposable projectives or the indecomposable injectives, the dimension types of the remaining projective or preinjective modules can be calculated either by the use of the
Coxeter transformation, or by using the additivity property of
dim
on exact sequences, here the Auslander-Reiten sequences.
Also, let us sketch the corresponding two-dimensional complexes
for some examples.
of type
E6:
The first example
F
discussed above was a quiver
188
Similarly, for the following quiver
F
of type
~6' we give the pre-
projective and the preinjective components of the Auslander-Reiten
quiver:
For the case of the quiver
F
discussed above of type
~6' the prepro-
jective and the preinjective components are
And finally, w e give the preprojective and the preinjective component
of one wild quiver:
t89
Note that in this case we can define inductively fixed irreducible
maps occurring in the Auslander-Reiten
to construct
the corresponding
sequences,
It should be noted that for a quiver
form
A-mp
with
preprojective,
P
indecomposable
those of the form
is of finite representation
both preprojective
with
module,
F
injec-
connected,
that
if and only if there is at least
and preinjective.
representations
nor preinjective,
there are no homomorphisms
indecomposable
It follows for
and preinjective,
The indecomposable
will always be called
I
type if and only if all modules are
one module which is both preprojective
ther preprojective
F, the modules of the
projective,
Aml
tive will be called preinjective.
kF
so that we are able
complex.
of a quiver which are nei-
are called regular.
Note that
from a regular module to a preproject~ve
and no homomorphisms
from a preinjective
jective or a regular module.
module to a prepro-
The shape of the category of representa-
tions of a connected quiver whidl is not of finite type therefore can
be remembered rather easily as follows:
reo iregar
I pren
the possible homomorphisms
suits concerning
An , Dn , E6 , E 7
going from left to right.
the regular part in the case of
or
~8' will be given in
F
A survey on rebeing of type
3.2.
For a proof of the results above, one can use the following
fact:
R = kF
for the path algebra
Reiten translations
A
A-
and
of the quiver
F, the Auslander-
in fact are functors,
and they have
the following properties:
(|)
and only if
As always:
X
if
X
is indecomposable,
is projective,
and if
X
then
AX = 0
if
is not projective,
X ~ A-AX.
(2)
Amx = 0
if
X
Therefore,
for some
m E~
by induction:
if
X
is indecomposable,
then
if and only if
X
is preprojective.
Thus,
is not preprojective,
then
X ~ A-mAmx
for all
m E ~.
190
(l ~)
X
Dually:
if
is injective, and if
(2 ~)
iff
X
if
X
X
X
is indecomposable, then
is preinjective.
X ~ AmA-mX
for all
(3)
If
is indecomposable, then
is not injective, then
Thus, if
X
A-X = O
iff
X ~ AA-X.
A-mx = O
for some
m E ~,
is not preinjective, then
m E~.
X
is without projective summand, then
Hom(X,Y) ~ Hom(AX,AY);
and if
Y
is without injective summand, then
Hom(X,Y) ~ Hom(A-X,A-Y).
Also,there is a linear transformation
c : ~n + ~n, called the
Coxeter transformations, with
(4)
If
X
is indecomposable and not projective, then
dim AX = c dim X, and if
X
is indecomposable and not injective, then
dim A-X = c I dim X.
The following lemma can be derived rather easily from
Lemma:
If
X,Y
(3):
are indecomposable and either both preprojec-
tive or both preinjective, then any homomorphism
f : X ÷ Y
is the
sum of compositions of fixed irreducible maps.
In particular, the endomorphism ring of any indecomposable
preprojective or preinjective module is just
k.
There are similar endofunctors, the so-called Coxeter functors
introduced by Bernstein-Gelfand and Ponomarev
[5 ], which are equi-
valent to the Auslander-Reiten translation functors up to categorical
equivalence.
Gabriel
The precise relation has been determined recently by
[20]. The Coxeter functors are built up from certain reflec-
tion functors which also will be used in the sequel.
For any vertex
from
r
£
r
of
F, denote by
OrF
the quiver obtained
by changing the orientation at all arrows involving
is a sink or a source, then there exists a functor
also denoted by
If
Or, and called a reflection functor, with the fol-
lowing properties:
Or(EF(r)) = O, and
the full subcategory of
summand
r.
MkF + MkorF ,
MkF
ar
is an equivalence between
of all representations without direct
EF(r), and the full subcategory
MkorF
of all representations
191
without direct summand
representation
of
E ° F(r).
•
F
wlth
r
Here,
(EF(r)) r = k.
out cycles, any change of orientation
of
Or, where we use only sinks
2.3.
EF(r)
denotes the simple
Note that for a quiver with-
can be achieved by a sequence
(or only sources).
Concealed Nuiver algebras
In this section, we will study those one-relation
algebras
R
which are rather similar to the path algebra of a (tame or wild) quiver
without relations,
"concealed"
and which can be thought of being "disguised"
forms of such path algebras.
or
The tame concealed quiver
algebras have the property that the deletion of any point immediately
leads to an algebra of finite representation
cuss other classes of one-relation
algebras by easy modifications,
quiver algebras,
type.
We later will dis-
algebras also obtained
but in contrast
to
these always will be proper enlargements
gebras of quivers.
Note that the representation
from quiver
the concealed
of path al-
theory of a concealed
algebra turns out to be rather similar to that of the corresponding
quiver.
Definition:
A finite dimensional
k-algebra
R
will be called
a concealed quiver algebra provided there exists a connected quiver
F
of infinite type with the same number of simple modules
there are cofinite
are equivalent.
subcategories
U
of
MF
and
In this case, we will say that
of the path algebra
kF, or just of
Let us consider
V
R
of
such that
MR
which
is a concealment
F.
some easy examples.
First, we show that we
can remove completely a zero relation of the following
... o
type
o
F :
where
F'
is a quiver with or without relations,
lowing situation
(with the same number of points):
o
o
o
o
o
and obtain the fol-
192
Namely,
if there is given a representation
~
~
~
with
a2
~,V
...
V2 . . . .
<
Vp
P
\
' , //
I
1 (
~
Vl
, ---+ V 2, ---+
-
82
8q
~ V
q
,
BI~I = O, then we form
Ker B q O ' ' ' o 8 1
<
Note t h a t u n d e r t h i s
+ ,., ÷ Ker 81 +'-- V 1 +-- V 2 ... +-EP Vp
1
functor,
we l o o s e p r e c i s e l y
~q(q+l)
indecompo1
and g a i n 2 - q ( 2 p + q + l ) , w h e r e a s , on t h e r e -
sable representations,
maining indecomposables,
the funetor
gories.
in case
then
V
T h i s shows t h a t
r
is a concealed quiver.
theorem
1, and
Next,
I-XX
consider
g i v e s an e q u i v a l e n c e
r'
is a quiver,
without
of subeaterelation,
This deals with the eases
of theorem
1-8
of
2.
the following
type of quivers with commutativity
relation:
0
C~
p
Pot
0
"
~r
2'
---
(q-l)'
0
.,.
oo.
1"
2"
Forming the pushout of the two maps
leads to a representation
p
p-1
0
0
• • •
q'
0
0
O--------nO
(r-l)"
a,~'
r"
for a given representation
of
2
1
0
C
_
1'
2'
v
~O
O
O
.. ,
l"
2"
~
~
(q-l) '
. • .
0
q'
0
4
O
~
(r-I)"
r"
The n e w quiver has the same number of points and no relation.
that we loose just one indecomposable
injective
one, but we gain
indecomposable
W I = WI, = 0
r+l
representations
and
W~ = k).
representation,
indecomposable
W
Note
namely a simple
representations (those
of the new quiver which have
Again, we see that the quiver with com-
193
mutativity
relation,
with cases
32-35;
was a concealed quiver.
in theorem
We will consider
cealments
In theorem
2, with cases
two algebras
of the following quiver
XXVIII-XXXI.
in greater detail,
F
I, this deals
of type
namely two con-
E7
O---~O---~O---~I~-'~)~=~D'~D
o
a quiver with a commutative
RI:
o
square
and
one with a zero relation
o
> o - - - ~ < ~ <
R2:
o----+o---->o---~~
o
In the case of
R;, forming the pushout of
between the full subcategory
Ker ~ N Ker ~' = O,
o o o
1
o
V1
of
~I
thus without direct
o o o, and the full subcategory
ations without direct
summands
e,~'
gives an equivalence
of all representations
UI
of
MkF
o o o I o o o
and
in case of
valence between
tions with
y
o o o o o o
o1 '
o
1
R2, forming the kernel
the full subcategory
surjective,
V2
thus without
of
direct
MR2
y
gives an equi-
of all representa-
summand of the form
~[KF
of all repre-
o o o o o 1 I
o
quivers:
and
o o o o o o I.
o
sentations
without direct
We compare
the three Auslander-Reiten
kF
R|
Y©Oi
U2
of
of
and the full subcategory
summand
of all represento o o o o o o.
I
Similarly,
with
surmnand of the form
194
R2
As in the case of a quiver, we see that the Auslander-Reiten
quiver separates into three different parts:
nent, the regular part
the preinjective
Len~a:
component.
Let
R
ment of the quiver
equivalent
modules
In general,
F.
Then
MR
F-modules.
subcategory
Proof:
in
Let
U
U, and
f : U ÷ U'
irreducible maps
in
MkF.
q : U ÷ V.
is irreducible
in
n(U) + ... ÷ ~(U')
ly, for
V,V'
in
V
with
R-modules
injective R-modules
F
a quiver,
Note that if
~F'
in
U
U' ,A U
(the
(the
cofinite
are in
then there is a chain of
V, since there can be at
n(f)
g : V ÷ V' irreducible,
-I
,)
(V
in U.
in
V; Similar-
there is a chain
n-|(V)+ ...÷ ~
in
preprojective
U, with Auslander-Reiten-sequence
o
also
V
U,U'
We want to show that for almost all indecomposable
representations
inde-
will be called preprojec-
most finitely many nontrivial ways of factoring
of irreducible maps
which is
will be called preinjective).
be cofinite
MR, with an equivalence
projective
PR
tive), the other containg the indecomposahle
IR
RR
The remaining
of the Auslander-Reiten
the indecomposable
in this subcategory
say a conceal-
has a full subcategory
form two components
one containing
in the corresponding
modules
compoand
we have:
be a concealed quiver algebra,
to the category of regular
composable R-modules
quivers:
the preprojective
(which is the union of many components),
belong to
U
-+ U ÷ U' + A U + o,
and
o + n(u) ÷ n(U') ÷ n(A U) + o
is an Auslander-Reiten-sequence
in
MR .
U
if
UI,U 2
has the following property:
jective,
and
f : U I + U2
Of course, we may assume that
is irreducible,
are indecomposable
then
U| 6 U
prepro-
implies
195
U 2 E U.
Then the first condition will be automatically
Now, there are only finitely many indecomposables
preprojective,
V ÷ ... ÷ X
where
÷ V
V E V
with
n-l(V)
such that there exists a chain of irreducible maps
X
is indecomposable
assume that we have choosen a minimal
V = V
satisfied.
. + ... ÷ V. ÷ X
q
q-i]
and since
~ (Vq)
and not in
V.
For, we may
chain
of irreducible maps, then all
• preprojective,
is
also
~-1(V I)
V i E V,
has to be pre-
projective.
Since there are only finitely many possibilities
and ~-I(VI)
is preprojective,
for
Vl
there can be only finitely many possi-
bilities for
V . Now if we choose V with ~-|(V)
preprojective
q
and such that for no chain of irreducible maps V ÷ ... ÷ X, the
module
X
lies again in
ting with
n-1(V)
Let
n
V, then the Auslander-Reiten
goes under
n
be the number of simple R-modules
We see that there are at least
form
A-mv
with
V E V
and
n
~-l(v)
with
Amv=
indecomposable
o.
X
with
V
is not
A-periodic,
preprojective,
It is also easy to see that for
V
there exists an
+ o, so there are all together only
modules
we must have
X
with
Amv=
o
Hom(X,V)
for some
~ o.
m ~ ~.
we see that the component of the Auslander-Reiten-quiver
the indecomposable
at least
n
R-modules
indecomposable
Similarly,
sable R-modules
V
with
~-I(V)
projective modules,
with
~-I(W)
posable injective modules,
preinjective
Since
Thus,
containing
preprojective,
contains
thus all of them.
we see that the component containing
W
V E V
m E
there are only finitely many indecompo-
Hom(X,V)
finitely many indecomposable
sequences of the
indecomposable
preprojective,
For, inside
sable modules
sequence.
(or simple kF-modules).
pairwise disjoint
~-|(V)
and none of them is periodic.
with
sequence star-
to an Auslander-Reiten
contains
the indecompoall indecom-
and also that this component differs from
the previous one.
It is now
rather easy to see that the regular
be supposed to be contained
R-modules.
in
This then finishes
kF-modules may
U, and that there are no additional
the proof.
Also, we should remark that it follows from the proof above
that for a concealed quiver algebra
preprojective
subcategory
tive subcategory
finite embeddings
R, say a concealment
PR' regular subcategory
IR, and similarly
of
kF, with
RR, and preinjec-
PF' RF' IF' there are full co-
196
PR U RR--~ Pp U Rp - ~ PR U RR
und
RR U t R - ~ RF U I p - +
Let us continue
quivers.
RR U I R .
to verify that certain algebras
are concealed
Consider the following
r
0
...
0
t'
0
(p-l)'
p'
s
we form the pushout
of
~,~'
in order to obtain representations
0
i'
..,
0
of
0
(p-I)' p'
f
s
Note that in this way, we loose just one indecomposable
and we obtain
W
p+q+2
n e w indecomposable
representations,
representation,
namely those
with support on the subquiver
] o
ro
and
with
|
o
I'
.°.
o
o
(p-i)' p'
W- = k.
Let us apply this in different situations.
If we start
r
R I with
q = I, and r' of type An, we obtain a quiver of
197
type
0
0 ... O
~
0 ... 0
thus a concealed quiver, and therefore
this deals with the cases
with
R2
with
38
we
and
XXXV.
F'
r
XXXIV.
An, we obtain the dual
consider the case of
q
1
b
>0-----+(~
If we start
R 2 is a concealed quiver,
can use induction and obtain after
>O
and
of type
RI, thus again
Finally,
itself is a concealed quiver:
36, 37, XXXIII,
q = 2, and again
of the situation of
R
0
. . .
0
R
where
this is
p = o, then
steps the quiver
q~-
q
>~(
0
i
S
Thus, if
F'
is a quiver without relations,
This is the situation of
71, with
F'
R
is a concealed quiver.
being of type
Dn, namely
• .. f>----Oa.
Next, consider
a •
R:
b
1
2
o
o
p-1
p
o
o
,..
di
Then, forming the kernel of
_
df
with
,/
~.b
1
o
e, we obtain a representation
2
o
...
p
P;-'--o
B a monomorphism ( t h e i n c l u s i o n map
a l l but
p+3
indeeomposable r e p r e s e n t a t i o n s
of the functor.
If
F'
is a concealed quiver,
quiver.
is of type
k e r a --+Ve).
of
9
R
Note t h a t
a r e i n the image
An, then we have seen that
thus, in this case, also
This deals with case
of
of theorem
R
is a concealed
I, and case
XXI
of
198
theorem
2.
Similarly,
we reduce
i
o
R:
o----o
1
o ... o
o
2
p
p-I
o
to
Ker ~ ~ o
R:
o
o ... o
1
forming the kernel of
monomorphism.
tions of
I0
and
o
p-1
p
and obtaining all representations
Note that for all but
R, the map
the cases
~
2
B
p+~
indecomposable
really is a monomorphism
.
with
B
a
representa-
This deals with
XXII.
Finally, we have to consider the case
83, namely
a
R:
p-1
Yp-I
We define a functor from
2 Y2 1
MR
to the representations
P"N
_
/p-2
'p-3
...
1
_/a
'0
p-1
as follows:
if
of the quivers
,,a
b
V
is a representation
of
R, let
V=I = {(x,y,z) C Va @ Vb @ Vi+ll~(x) + ~(Y) + 71'''Yi (z) = O}
with the obvious maps.
In this way, one can show that
R
is a con-
A
199
A.
cealment of
2.4.
Vectorspace
categories
One of the main working tools in the sequel will be the notion
of a vectorspace
category,
and, as a special case, the additive cate-
gory of a partially ordered set.
Nazarova and Roiter
a rather eleborate
present,
[28,30],
theory.
These concepts were introduced by
and they and their students developped
We will use many of their results.
At
the criteria for partially ordered sets to be of finite or
tame representation
type seem to be the most powerful tool in repre-
sentation theory.
By definition,
a
a vectorspace
additive k-category
category
K
is nothing else than
together with a faithful functor from
K
to
the category of k-vectorspaces
(usually denoted by
I'l)
idempotents
X
K , we will call
in
K
the vectorspace
split.
IXI
If
is an object of
the "underlying vectorspace"
the forget functor.
Recall that the condition
2
split, means that given an idempotent
e = e
of an object
that
e
Using the forget functor
in the endomorphism
l'i, a vectorspace
(such as a fixed subspace,
K(X,Y)
Hom(JXJ,IYI)
K(X,Y)
exists the direct sum
sumed),
X
of
K
may be
from
X
X
(not further specified)
a fixed endomorphism etc), and
to
Y
form a k-linear
structure.
X @ Y
Note that we assume that
in
K.
Also,
is finite-dimensional
then our assumption
subspace of
- the elements
of as those linear transformations
which means that for any two objects
vectorspace
category
of all linear transformations
may be thought
preserve the additional
additive,
such
Thus, there are given objects which may be thought of
the homomorphisms
of
X = X| @ X 2
K
ring
of the category of
consisting of a vectorspace with some additional
the space
I'I
in
X I.
considered as a (usually not full) subcategory
k-vectorspaces.
X, and
that idempotents
X, there exists a direct decomposition
is the projection onto
structure
of
such that
X, Y
in
which
K
is
K, there
in case the underlying
(and this usually will be as-
that all idempotents
split implies that
can be written as the direct sum of a finite number od indecomposable
objects and these have local endomorphism
position
is unique up to isomorphism.
rings.
Thus,
such a decom-
200
The usual construction
use of functors
such as
of vectorspace
Hom
or
Ext I.
ditive k-category with split idempotents,
we denote by
the form
Hom(M,K)
the vectorspace
Hom(M,X), with
X
categories will be by the
Namely,
and M
let
K
be any ad-
an object in
K. Then
category whose objects are of
an object in
K, and with maps
Hom(M,X) ------+ Hom(M,Y)
being of the form
Hom(M,f),
we consider something
functor
Hom(M,-).
lying vectorspace
space categories
dule category
Hom(M,X).
such as
Ext i
Given a vectorspace
U
is given by a pair
with
A homomorphism
(~,B)
where
(k,o,O)
where
U(K),
If
(U,~,X)
U(K)
from
Clearly,
(U,~,X)
IXI
to
is k-linear,
U(K)
is
(U,~,X),
K, and ~ : U ÷
is a
(U',~',X')
B : X ÷ X'
is again an addi-
the direct sum of
(U,~,X)
Clearly,
to a direct sum of a triple
denotes the zero object of
K.
and
any
(U,~,X)
This explains the
up to the one indecomposable
object
we may suppose that we deal with an object of
preassigned
in the mo-
being an inclusion map and several copies of
notion subspace category:
in
is a module
(U @ U', (~
0 ) X O X ,) .
\0 ~,~,
is isomorphic
O
M
~ : U ÷ U'
B~ = ~'~.
being given by
~ : U + IXI
where
is an object of
K, such that
U(K)
Thus
be just the under-
K, its subspace category
tive category with split idempotents,
object of
K.
is defined).
category
X
k-linear transformation.
(U',~',X')
IHom(M,X) I
its objects are triples of the form
is a k-space,
is a map in
is a map in
In a similar way, we define vector-
Exti(M,M),
(so that
defined as follows:
where
f : X + Y
Of course, we let
of
M
where
like the image category of the representable
subspace
(note:
belongs to
a subspace,
U(K),
then
K
not necessarily
dim U + dim IxI
(k,o,O)
with a
a subobject!).
is called its
dimension.
As in the case of algebras,
have to distinguish
category
K
is
K
wild
the different
also for vectorspace
representation
is of finite representation
t~pe,
if there is an exact embedding of
is a representation
categories we
types.
if
M~
A vectorspace
U(K)
is finite and
into
equivalence with the image category.
U(K)
which
We have to
be slightly more carefully in order to define the notions tame and
domestic:
then
K
In case
K
has only finitely many indecomposable
is called tame provided
for any dimension
finite number of embedding functors
objects,
d, there is a
F i : Mk[T] *U(K)
such that all
201
but a finite number of indecomposable
are of the form
this case,
Fi
K
Fi(M)
objects of
for some indecomposable
U(K)
of dimension
k[T]-module
M.
d
In
is called domestic provided we can choose the functors
independently
from
d.
In general,
a vectorspace
category
K
is
called tame or domestic provided any full subcategory with only finitely many indecomposable
objects is tame, or domestic,
respectively.
Of great interest is the special case of a vectorspace
K
where for any indecomposable
End(X)
object
X
of
is a division ring (thus, in case
equal to
k).
In this case, K
if all indecomposable
k
is algebraically
is called Schurian.
objects of
K
category
K~ the endomorphism ring
closed,
In particular,
are one-dimensional,
then
K
is
Schurian.
Lemma:
Let
k
Schurian vectorspace
be an algebraically
category.
If
K
closed field, and
type, then any indecomposable
object of
is not of wild representation
type, then any indecomposable
K
is at most two-dimensional.
indecomposable
Hom(Y,X)
in
K, and
K
be a
is of finite representation
K
Moreover,
is one-dimensional.
in this case, if
dimlX I = 2, then
Hom(X,Y)
t 0
If
K
object of
X, Y
are
or
+ O.
Proof:
Denote by
the direct s u ~ a n d s
category with
U(add X)
of
add X
X°
the additive subcategory
Now let
End(X) = k, and with
X
generated by
be any object of a vectorspace
dimIX I = n.
Then clearly
is isomorphic to the category of representations
of the
quiver
with
n
a~ows,
thus
add X
is of finite representation
n = 1, and is of wild representation
X,Y
then
are objects with
U(add
X • Y)
type for
End(X) = End(Y) = k
is isomorphic
and
n ~ 3.
type only for
Similarly,
Hom(X,Y)
if
= O = Hom(Y,X)
to the category of representations
of
the quiver
where
n = dimIXl,
m = dimlY I.
This category
is wild in case n+m >_ 3.
202
The vectorspace categories with only one-dimensional indecomposable objects have been investigated in great detail by Nazarova and
Roiter and their students.
Namely, these correspond just to the parti-
ally ordered sets.
be a partially ordered set.
k-category
for
kS
Let
S
as follows:
Define a
its objects are the elements of
S, and
i,j C S, let
k
i ~j
Hom(i,j) =
iff
,
0
ilj
with composition of maps the ordinary multiplication in
k.
In order
to deal with an additive category, we just adjoin the finite direct
sums, and obtain in this way the category
space category if we consider
any element
i
of
S.
k
We will call
the partially ordered set
S.
sional indecomposable objects.
add kS.
This is a vector-
as the underlying vectorspace for
add kS
Note that
the additive category of
add kS
has only one-dimen-
Al~o, conversely, assume
K
is a
vectorspace category with only one-dimensional indecomposable objects
(and such that all objects are finite-dimensional).
is of the form
add kS.
We claim that
Namely, the indecomposable objects in
better their isomorphism classes) form the elements of
X,Y
indecomposable in
j = [Y], we have
i j j
K, with isomorphism classes
iff
Hom(X,Y) # O.
K
K
(or
S, and for
! = [X]
and
Clearly, in this way we
obtain a reflexive relation which is both transitive and anti-symmetric
due to the fact that the indecomposables are one-dimensional.
There is another interpretation of the subspace category
U(add kS)
of the additive category of a partially o~rdered set
definition, an S-space
(V,V i)
subspaces
i C S, such that
Vi, for all
V i c Vj.
to
The S-spaces
(U',U~)
fying
form a category
i J j
V
in
U(add kS)
equivalent where
S°p
lying set of
Namely, let
S.
i E S, let
all images
f : U ÷ U'
we denote by
and
S(S °p)
(U,Ui)
satis-
[17], and he
are representation
is the reversed partially ordering on the under-
Xi
U i ~ Uj;
q(U,~,X).
(U,~,X)
be an object in
be the subobject of
i ÷ X, and let
X i ~ Xj, thus also
implies
S(S), with maps from
being given by linear transformations
By
together with
S
f(Ui) _c U~.l This definition is due to P. Gabriel
showed that the categories
For any
is a vectorspace
S.
X
U i = ~-l(xi).
For
therefore
(U,U i)
U(add kS).
which is the union of
i E j, we have
is an
s°P-space which
203
Proposition
(Gabriel)
dense functor which reflects
:
n:
(add kS) + S(S °p)
isomorphisms,
is a full,
thus a representation
equi-
valence.
In fact, one shows that the objects
covers,
of
and that
S(s°P),
U(add kS)
see
[18]
In contrast
or
S(S °p)
to this subspace interpretation
U(add kS)
have projective
covers
[15].
dealing with one vectorspace
definition of
in
is just the category of projective
of
U(add kS), of
with a set of prescribed
subspaces,
may seem to be less intuitive.
the
However,
in
the sequel, we always will need to work with this definition directly.
Namely, we usually will have to consider
form
U(Hom(M,M)),
just the additive
and sometimes
category of a partially ordered set.
should remark that the definition of
free translation
subspace categories
it will turn out that
U(add kS)
of a very important
has been considered
k
is
Also, we
is precisely
the base-
class of matrix problems which
for a long time by Nazarova and Roiter.
there are given matrices over
of the
Hom(M,M)
Namely,
with the columns labelled by ele-
ments of a partially ordered set, and the rules of allowed transformations are as follows:
tions
we are allowed to use all possible
row opera-
(adding multiples of one row to any other row, and multiplying
any row with a non-zero scalar), we also may multiply any row with a
non-zero
scalar, but we only may add a multiple
to a row labelled
t, if
s < t
In order to visualize
in
the elements
sented by points,
~
and the relation
a2
chain
disjoint union of
set
S
will be repre-
For example,
b2
{al,a2,b],b 2}
{1 < 2 < ... < n}
(n]) . . . .
of
is always thought of as going
from left to right along the drawn edges.
is the four-element
s
partially ordered sets rather easily, we
will use the following convention:
The n-element
of a row labelled
S.
(ns)
by
with
a i ! bj
for all
will be denoted by
(n I ..... ns).
denote the following partially ordered set
By
N
i,j.
(n), the
we will
204
and
(N, n2,...,ns)
(n2,...,ns).
stands for the disjoint union of
(The reason for drawing the relation
right, and not as usually from below upwards,
N
!
and chains
from left to
comes from the fact that
most partially ordered sets considered will be derived from AuslanderReiten-quivers,
the relation
~
being derived from the arrows
which usually are drawn from left to right.
tially ordered set as a vectorspace
category,
Also considering
the relation
nothing else then the existence of a non-trivial
map
J
the parmeans
~ .)
Let us quote now the two main theorems on the representation
type of partially ordered sets.
partially ordered set
S
By the representation
type of the
we will mean that of the vectorspace
category
add kS.
Kleiner's
representation
theorem.
The partially ordered set
type if and only if
as a full subset one of the sets
or
S
S
is of finite
is finite and does not contain
(I,I,I,1),
(2,2,2),
(1,3,3),
(1,2,5)
(N,4).
This theorem has been proved in
Nazarova and Roiter developped
in
[22]
[28].
main working tool, the "differentiation
ments", has been given by Gabriel in
using techniques due to
A different
with respect
Let us remark that Kleiner has given a complete
list of all partially
type which have a faithful repre-
this result will not be needed here, for the one-relation
algebras,but
we will comment on it further in the report on the Brauer-
Thrall conjectures
Nazarova's
representation
of the sets
(N,5).
to maximal ele-
[18], using homological notions.
ordered sets of finite representation
sentation:
approach to the
Also,
[35].
theorem.
The , partially ordered set
type if and only if
(1,1,1,1,1),
if
S
S
(1,1,1,2),
(2,2,3),
is of wild
1,2,6)
type, then
or
S
type.
The proof of this theorem is in
Let us also mention
(1,3,4),
is not of wild representation
is of tame representation
S
contains as a full subset one
that Otra~evskaja
[27].
has given a complete classifica-
205
tion of the partially ordered sets with precisely one series of indecomposable representations.
Let us return to the general vectorspace
not necessarily
obtained from a partially ordered set.
adopt conventions
for visualizations,
The case of a non-Schurian
The one-dimensional
•
K
in part
category of finite representa5.
Here, we will deal with a
which is not of wild representation
objects of
K
, the two-dimensional
type.
will again be represented
by ver-
objects either by black squares
or, if we want to specify two base vectors,
vertices.
We also will
at least for some special cases.
vectorspace
tions type will be considered
Schurian vectorspace
tices
categories which are
by squares
~
Edges will represent non-zero homomorphisms,
~,
with two
but usually we
will have to specify rather carefully the relations.
Two special situations will occur more frequently,
seems to be helpful to introduce
the following convention:
so that it
We will
use the two symbols
and
in the following
with
8
situation:
one-dimensional
vial endomorphism
satisfying
The first refers to a vectorspace
objects,
I
two-dimensional
category
object with tri-
ring, and non-zero homomorphisms
the following relations:
3
3
Z aiB i = O, Z Yi~i = O,
i=l
i=l
This implies that the images of
rent one-dimensional
Yl' Y2' Y3
are three pairwise diffe-
subspaces of the two-dimensional
are also just the kernels of
ber of veetorspace
Biy I = B2Y 2 = 63y 3 = O.
61' ~2' 63'
subcategories
object,
respectively.
and these
A great num-
of this form will be exhibited
in
206
section
3.4.
In
3.3, one special case will be derived in great de-
tail.
%
The second s)~mbol
4
one-dimensional
endomorphism
where
in
P
~
refers to a vectorspace
objects,
I
two-dimensional
category with
object with trivial
ring, and non-zero homomorphisms
Examples of this type will also be found
~iB2 = O = ~2B|.
3.4.
2.5.
Let
Subspace categories
R
arising from simple injective modules
be a finite dimensional
injective R-module with endomorphism
factor ring of
tive cover of
R
E.
without
Lemma
category
|:
M(kMs)
Recall,
M(TM S)
)
)
M
is
MR
v
Let
E
a simple
be the largest
P(E)
a projec-
the radical of
S,T
is equivalent
of the bimodule
and a bimodule
TMs
P(E);
(~,B)
of maps
By = y'(~ @ |M ).
as the category of triples
to the
kMs.
TMs , the category
has as objects the
: U T ~ TMs ÷ X S), and a map from
satisfying
S
E, and
of R-modules
of the bimodule
is given by a pair
B : XS ÷ XS
described
M = rad P(E)
of representations
that for rings
(UT,Xs,Y
)
(UT,Xs, Y )
k.
factor
and
an S-module.
The category
of representations
triples
ring
composition
We denote by
it is easy to see that
k-algebra,
(UT,Xs,T)
to
~ : U T ÷ U~,
This category also can be
(UT,Xs, Y : U T ÷ Hom(TMs,Xs)) ,
~
using the adjoint
y
of
y.
For the proof of lemma
Let
since
e
be an idempotent of
E = eR/rad(eR)
Also, M = eR(l-e).
l, we may suppose that
R
with
is injective,
We can write
R
P(E) = eR.
and
Then
R
is basic.
(1-e)Re = O,
S = (|-e)R(l-e)
= (|-e)R.
in matrix form as follows:
207
R =
I
eRe
eR(l-e)
1
e
and the representations of such a matrix ring can be described by
triples
(Uk,Xs, U ~ M S ÷ XS).
If we start with a quiver with relations
(F'Pi)i E I' and
R = kP/<Pili E I>, and we consider the simple R-module
responding to the vertex
is a source.
from
Let
(F'Pi)i E I
subquiver of
a
P
a, then
(Fa'Pj)j E J
R
is injective if and only if
by deleting the vertex
Then
Fa
MR
is the full
S = kFa/<Oil j E J>
E, and
{Pili E I}.
is the largest factor
MS
is the full sub-
of all R-modules without composition factor
Given a module
of all objects
XS
MS, let
with
a
a, and, if
in fact is a subset of
without composition factor
category of
a.
containing all vertices different from
Note that in this situation
cor-
be the quiver with relations obtained
is a source, then {@jlJ e J}
ring of
E
E = E(a)
r(M S)
E.
be the full subcategory of
Hom(Ms,X S) = O.
Recall that
MS
Hom(Ms,M S)
de-
notes the vectorspace category given by the representable functor
Hom(Ms,-).
Lemma
subcategory of
2:
Let
S
M(kM S)
be a finite dimensional k-algebra.
of all representations of
zero direct summand of the form
sentation equivalent to
Proof:
r(M S)
(O,Y,o)
with
U(Hom(Ms,Ms)) , where
Clearly, the objects of the form
generate the kernel of this functor
kMs
Y
in
The full
without nonr(Ms), is repre-
(Uk,Xs, Y)
(O,Y,o)
(Uk,Xs,Y)~-+~.
goes to
~.
with
in
Y
On the
other hand, the functor is full and dense, and it reflects isomorphisms
if we restrict it to the subcategory of representations without nonzero direct summands of the form
(O,Y,o), with
Y
in
r(Ms).
According to the le~m~a, the representation type of
both on
r(Ms)
and
U(Hom(Ms,Ms)).
MR
depends
It is easy to construct examples
where they have completely different representation types:
of them may be wild, the other of finite type.
any one
However, since most
considerations will be done by induction on the number of simple modules, thus the number of vertices of
r(M S)
P, we then may assume that
is known and will have to concentrate on the subspace category
208
U (Hom(Ms, M S) ).
In the case of one-relation algebras
the s t a r t i n g p o i n t of the r e l a t i o n
a
in order to construct
advantage that
Fa
S
and
R = kF/<O>, w i t h
U(Hom(Ms,Ms)).
is a quiver w i t h o u t relations,
means that we k n o w the S-modules rather well.
a
does not need to be a source,
a
being
0, we u s u a l l y will take this v e r t e x
This clearly has the
thus
S = kFa, w h i c h
Of course,
in general
so that we cannot apply the c o n s i d e -
rations above,
However, we see that most of the algebras listed in
theorem
1.5
F
I
of
decomposes into
have the following property:
P = F' U F"
w h i c h have only the v e r t e x
in
O
all lie inside
Clearly,
a
where
F',F"
in common,
F', a
are two full subquivers
such that the paths involved
is a source in
F', and
F"
is a tree.
in this case we may apply r e f l e c t i o n functors to
respect to some points lying inside
source also in
F".
For example,
a
b
F'"-{a}
so that
a
MF
with
becomes a
in the case of
c
dfoe
g~-~oh
~-~o--~o
r
we f i r s t
apply
order
obtain
to
s
reflection
t
functors
a
b
~c~'o
at
the
points
c,b,
and
c
d~--oe
g~-~oh
~ , ~ o---+o ,
r
then
ra
has the f o r m
rT
a
o+---o
~
s
t
F a = F'a 0 P"a
F"
a
o--+o
~o
~o-+o
and
M
is the r e p r e s e n t a t i o n of
Fa
w i t h d i m e n s i o n type
c,
in
209
I+--o
1
1---+t
1--+1
o-'-+cy-+o
Now
F a' is a quiver of type
jective,
thus
r(M)
contains
E7, thus tame, and
all
preprojective
M' = MIF'a
and a l l
is prein-
regular
r'-modules.
a
We show that
Hom(M',MkF,)
a
Namely, we determine
is of the form
the position of
1 o
1
1 1
in the preinjective
o o o
component o f
MkF,
a
M'
and encircle those indecomposable
kF'-modules
X with Hom(M',X) # o.
a
This is rather easy, since any non-zero map M' + X has to be a sum
of compositions
of
M'
of irreducible maps, thus only the modules to the right
may occur.
dim Hom(M',X)
= I
The same calculation
Of course we know that for
gory
Hom(M",MkF,,)
is of the form
a
thus
Hom(M,MkF )
shows that
in case it is non-zero.
is
M" = MIF a , the vectorspace
cate-
210
which is of tame type according to the theorem of Nazarova, but not
of finite representation type, according to the theorem of Kleiner.
In fact, H o m ( M , ~ F)
contains a full subset of the form
Let us write down
M
(F,p)
(2,2,2).
also in some other cases:
Fa
\
dim M
/
o
o
1
1
2
1
'~
I
1
1
1
Our further interest will usually be concentrated around the
calculation of the vectorspace category
the path algebra of a quiver
A
Hom(Ms,MS) , where
In order to know whether
Hom(Ms,M S)
is Schurian or not, we
have to consider the endomorphism rings of the objects
Hom(Ms,Ms).
I
But clearly this endomorphism ring is
is the annihilator of the
End(Xs)-module
if for all indecomposable modules
End(X S) = k, then
pens in case
MS
Hom(Ms,Ms)
S = kA,
(without relations).
XS
with
is Schurian.
is preinjective.
Hom(Ms,X S)
End(Xs)/I
in
where
End(X¢)Hom(Ms,Xs).
Thus,
H°m(Ms~X S) t o, we have
In particular, this hap-
For, in this case
H°m(Ms,X S) t o
211
implies for
therefore
XS
End(X S) = k.
Lemma
MS
indecomposable that also
2:
If
S
XS
is preinjective, and
Thus we have shown the following:
is the path algebra of a connected quiver, and
is preinjective, then
Hom(Ms,M S)
is Schurian.
Also, we note some necessary conditions on
MS
for
Hom(Ms,M S)
to be of finite or tame representation type.
Lemma
and
MS
If
Let
Hom(Ms,Ms)
Hom(Ms,M S)
If
3:
S
be the path algebra of a connected quiver
A,
a module.
is a finite category (in particular, if
is of finite representation type), then
Hom(Ms,Ms)
MS
is preinjecti~.
is of tame representation type, then either
is of finite representation type, or
MS
A
has no non-zero preprojective
direct summand.
Proof:
For both assertions, we may suppose that
MS
is indecom-
posable.
If
Now
M
is not preinjective, then
H°m(A-mM, I) t 0
AmA-mM ~ M
for all
m E~.
for some indecomposable injective module
I, and
therefore also
Hom(M,Aml) ~ Hom(AmA-mM,Aml) ~ Hom(A-mM, l) + O.
Thus
Hom(M,Ms)
Now let
has infinitely many indecomposable objects.
M
be preprojective, say
posable projective module
P(r).
representation type, and
I
the dimension of
In case of
A
for some indecom-
being not of finite
indecomposable injective, we know that
dim Hom(P(r),AnI) = dim(AnI)r
bounded.
M =Amp(r)
is unbounded
(for
n + ~), thus also
Hom(M,AnI) = H o m ( A - m p , A n l ) ~ Hom(P,Am+nI)
However,
for
Hom(Ms,Ms)
have seen that the objects
dim Hom(Ms,X S) ~ 2.
XS
with
is un-
of tame representation type, we
End(X S) = k
satisfy
212
2.6.
Finite enlargements
We consider now algebras
We always will consider
fying the S-module
MS
XS
a finite enlargement of
Note that in case
S
Lemma
with the tripel
(O,X,o).
is cofinite in
in case
MS
TMs .
M(TMs) , identi-
We call
M(TM S)
M(TMs).
is the path algebra of a connected quiver, this
Let
l:
together with a bimodule
MS
immediately implies that
ducible, and
S,T
as a full subcategory of
MS
TMs
is preinjective.
be a bimodule.
Hom(Ms,Xs) = O.
Let
f : XS ÷ YS
be irre-
Then also (o,f) : (O,X,o) ÷ (O,Y,o)
is
irreducible.
Proof:
Let
(o,f) = (gl,g2)(hl,h 2)
be a factorisation, with
(hl,h 2) : (O,Xs,o) --+ (UT,Zs, y : U T 0 TMs ---+ ZS).
Now
h2
f = g2h2
is a factorisation of an irreducible map, thus either
is a split monomorphism or
g2
a split epimorphism, then also
assume
h2 : X S ÷ ZS
CS, and
If
is a split epimorphism.
is a split monomorphism,
for some complement
zero homomorphism
is a split epimorphism.
(gl,g2)
thus
g2
is
So
Z S = h2(X S) @ C S
y(U T O TMs ) c CS, since there is no non-
M S ÷ X S ~ h2(Xs).
As a consequence,
(hl,h 2)
is a
split monomorphism.
Recall that we identify M S
namely we identify
Corollary:
and
TMs
XS
Let
of
S
MS
with a full subcategory of
with
(O,X,o)
in
M(TMs) ,
M(TMs).
be the path algebra of a connected quiver,
be a bimodule with
MS
preinjective.
Then all components
but the preinjective one of the Auslander-Reiten quiver of
unchanged in the Auslander-Reiten quiver of
MS
remain
M(TMs).
This follows directly from the previous lemma, since the projective S-modules remain projective in
M(TMs) , and the Auslander-Reiten
sequences
f
O--+ XS --+ YS ~
in
MS
ZS --+ o
which do not involve preinjective modules remain Auslander-
Reiten sequences in
M(TMs), since
Hom(Ms,X S) = O = Hom(Ms,Y S)
the sequence is characterised by the fact that both
irreducible.
f
and
g
and
are
213
Of course, we similarly
tive component of
MS
is changed which involves modules
chains of irreducible maps
sable direct summand of
Lemma
TMs
2:
Let
M' ÷ ... ÷ X, where
S
be the path algebra of a connected quiver,
MS
cofinite
representation
a chain of irreducible maps
indecomposable
Proof:
of
M'
in
TMs
M(TMs).
with
If
(UT,Xs,Y)
let
is an
y ~ o, then there exists
M' + ... + (U,X,y),
direct summand of
Let
X
with
S
is an indecompo-
M'
MS.
be a bimodule with
indecomposable
see that only that part of the preinjec-
where
M'
is an
M S.
be an indecomposable
direct summand of
M
such
AP
that
yIU ® M'
with
~
is non-zero.
a free module.
Then
w
direct sum of copies of
Let
o t y' : M S + X S
Also,
MS
let
e : U ÷ U
y(e 0 IM,)
to
be an epimorphism
is a non-zero map from a
XS, thus we see that
and consider
(o,y')
H°m(Ms,X S) t o-
: (O,M,o) ÷ (UT,Xs,Y)-
Note that there are only a finite number of indecomposable
triples in
M(TM S)
the indecom-
which have a non-zero map from
posable preinjective
the remaining
consequence,
S-modules YS
indecomposable
with
(O,M,o)
:
namely,
Hom(Ms,Y S) /+ o, and some of
triples which do not belong to
the process of factorizing
(o,y')
MS.
non-trivially
As a
has to
stop after a finite number of steps.
As a consequence,
M(TM S)
is obtained
we see that the Auslander-Reiten
from the Auslander-Reiten
the preinjective
component
Auslander-Reiten
quiver of
MS
exists a chain of irreducible maps
M(TMs)
MS
by extending
we know that the
will have the shape
where we have marked the indecomposable
Reiten quiver of
quiver
in the following way:
quiver of
modules
XS
M S + ... ÷ X S.
for which there
The Auslander-
is then of the following form
214
where the additional
region right of
indecomposable
modules
lie in the
M, and most of them (at least those with
have a chain of irreducible maps
y ~ o)
M + ... ÷ (U,X,y),
We consider one example in detail:
Here, S
(U,X,y)
the one-relation
algebra
is the path algebra of
0
the module
MS
is of dimension
O
type
I
1
0
1
I
|
, and
r
=
!
The preinjectives
of
MS
form the following component:
Note that we can calculate without difficulty
that
the additive category generated by the partially
Iom(Ms,M S)
ordered set
is
k.
215
The component of
M(kM S)
containing
M
has the following form
and we have encircled the indecomposable modu!es belonging to
M S-
Note that this component contains all nine indecomposable injective
modules, and one indecomposable projective module.
2.7
Gluing of two components
We proceed now to consider bimodules
no longer a finite enlargement of
injective module
finite type.
TMs
for which
MS, but where again
MS
M(TM S)
over the path algebra of a connected quiver of in-
It turns out that in this situation we also have to
work with the dual situation of the path algebra kS' of a connected
quiver of infite type and a preprojective S'-module
consider the following one-relation algebra
R
NS,.
Namely,
given by
Of course, our previous device was to consider the subquivers
g:
o+--o+--~
and the indecomposable
and
TMs
T:
of dimension type
is
is a pre-
216
I1
so that
MR
is
,)
1 I
dim M S =
,
dim T M = (9 0 O)
I I l I I
M(TMs).
reduced to
However, we also may consider
the subquivers
T':
~--o
and
and the indecomposable bimodule
dim T,N = (0 0 9),
T,Ns,
of dimension type
dim NS, =
(
1
It is easy to see that
)
I I I
1
I
I
1
M R can be reduced to the category of triples
(Xs''YT''Y : XS' --+ YT' 0 T,Ns,), thus to the category
Of course, we always will consider
ries of
MR .
Note that
NS,
MS
and
MS,
and
M((T,Ns,)#)
where
quivers of infinite type, M S
with
S,S'
MR.
Hom(MR,N R) ~ o.
M R can be reduced both to
are path algebras of connected
is preinjective, N S,
Hom(b~,N R) ~ o, and, moreover, M S U M S ,
gory of
M((T,Ns,)~).
as full subcatego-
is preprojective and
In the following, we will assume that
M(TM s)
.
is preprojective
is a eofinite subeate-
Note that these conditions all are satisfied for the
one-relation algebras listed in theorem
l
and marked with
"G".
This letter stands for gluing since we will see that the Auslander-Reiten quiver of
and
MS,
MR
is obtained from the Auslander-Reiten quivers
by joining together the preinjective component of
the preprojective component of
MS
MS
and
MS,.
Let us first show that there is a chain of irreducible maps
MR+
... ÷ N R.
: MR+
it
N R.
By assumption, there exists a non-zero homomorphism
Let
G
be the following set of indecomposable R-modules:
should contain the indecomposable S-modules
H°m(Ms,X S) t o, the indecomposable S'-modules
X
with
YS'
with
Hom(Ys,,Ns,) ~ o, and those indecomposable R-modules which are not in
M S U MS,.
Then
G
has only finitely many objects, and any non-tri-
vial factorisation of
y
a direct sum of modules in
torising
y
will be of the form
G.
MR+
GR ÷ N R
with
GR
As a consequence, the process of fac-
non-trivially stops after a finite number of steps, thus
217
there exists a chain of irreducible
As a consequence,
MR
outside of
... + N R.
we see that the Auslander-Reiten
S'-modules and the indecomposable
M S U MS,.
All other components
and are unchanged.
all components
MR.
MR+
has one component which contains the preinjective
preprojective
MS,
maps
of
Similarly,
MS
in
MR .
of
MS,
l
of
X S ÷ ... ÷ M S
similarly,
remain chains of irreducible maps in
the chains of irreducible maps
there is a chain of irreducible maps
R-modules
XS
and
YS'
(some preinjective
and the indecomposable
S-modules,
of
2.6
of the Aus-
some preprojective
M S U MS, ) .
S'-modules,
Since there can-
we conclude that these additional
also have to belong to the component
application
Since
There are only finitely many additional
R-modules outside
not be a finite component,
MR .
MR ÷ ... ÷ NR, we see that all
belong to the same component
MR .
in
one,
2.6, we know that
MR;
quiver of
2.6, that
remain chains of irre-
ducible maps in
lander-Reiten
or in
one, remain unchanged
NS' ÷ "'" + YS'
these modules
MS
but the preprojective
Again from lemma
the chains of irreducible maps
the
R-modules which are
lie either in
For, we know from the Corollary in
but the preinjective
all components
remain unchanged
quiver of
S-modules,
containing
MR
shows that the Auslander-Reiten
and
R-modules
N R.
A final
sequences
o ÷ X I ÷ X2 ÷ X3 ÷ o
in
MS
with
quences in
Hom(X3,N)
Hom(Xi,M)
= Hom(X2,M)
MR, and similarly,
= o
= o
those in
remain Auslander-Reiten
Auslander-Reiten
quiver of
MR
remain Ausiander-Reiten
MS ,
with
sequences
is obtained
Hom(X2,N)
in
MR .
from those of
MS ,
L!
as follows:
(
M
S
~
_
NS'
se-
=
Thus, the
MS
and
218
\/\/\~/\
/\/\/t\
/
\/\
/ ~r,/\
ooo-o~ooooo
\ /\
oo~"~°°°ooo
\
o0@"~%oo
~
o
/t\/
~
~c
\ /\t/\
~o""~°%oo
oo..."-:" ooo ('~"o o----:.--ooo
\/\
\~!
/,,,,
/ \ / \
\ / \ /
O. - ° ~ - % . c , 0
/\/\
i.oO~°°o.
000°%-°.°
\/\
ooo--:-oo. ,~o~
,-
,o.--
"--oo
~ . . . T . ~,~%.
o
~
\/
ooo°°~.-'.oo
o0o" "...",-~.o
ooo"~"..-.-
/\/~\/
/\/\t/\
/'\/'\/t\ /
\
/\/\t/\
.i
o
o
219
The new component of
~
containing both modules
MR
and
NR
will
be called the glued co~onent.
For exa~le
in case of the one-relation algebra
R
given by
t h e g l u e d component i s e x h i b i t e d on t h e p r e v i o u s p a g e , I n g e n e r a l ,
c a s e we d e a l w i t h t h e
in
(usually wiid) quiver
with all possible c o ~ t a t i v i t y
relations, let
S
be the path algebra
of
...
~
...
O
0
0
and
S'
the path a l g e b r a of
Then t h e p r e i n j e c t i v e
S - m o d u l e s and t h e p r e p r o j e c t i v e
c o m p o n e n t s o f t h e f o l l o w i n g form:
7\
/
/
/
S ' - m o d u l e s form
220
and the glued component of
J~R
J
i //
looks as follows:
i
1
As a n o t h e r
relation
Here,
e x a m p l e , we e x h i b i t
the glued component of the one-
algebra
S
and
S'
and t h e d i m e n s i o n
a r e g i v e n by t h e f o I I o w i n g
type of
MS i s
component of
MS i s
..~14
, that
quivers
of
NS, i s
t
t4
to
t
preinjective
the preprojective component of
'
MS,
is
The next page shows the glued component of
M R.
• The
221
~"
~
~~oo
'~
\ / \T/\
/
o/ \ /t\
/ \
e~
\./
~
\~,/ \
~0,...~
°
~
O0 O0
,-~\
\_/
" ~ o "°°
/t\
\J/
".-o--'%° , - ' ; ° % 0
..
o
/
\
~.o ,- o
-~Z3 -°o
/\
\_/
. . . . . oo
/\
\/
"~"~o
O0
.~.-~..
rl..~
o/ \ /J\
/ \ / \
\o/
\ J / \ / \ /
o/ \o/~\
,7 \ / \
\o/ \ t/ \/
\ /
0 °~° 0° % 0
.--..-'.'oo o o o ' 0 o . , "~~ ' - , - o
0 Oooo o o
° °o° °- % °
~ ¢~r~.O
. - -. -. - . o
o -.% - . o
0%;-.0
0
"""'..
~
/\
\o/
/ \
\/
-.~"%, . o ' - ' . . . ,
o-"-o..
"'~-~'~
0 ~rl~.
o"~"*'"%~ ~
o0~"''~o
\/T\
/\I/
\/t\
0o%%0
0
o OoToO 0 o~
o
0o0-;%0
0%%-00
mg;-.o
-
0 0.ooo
o
O0
'~'~"
;~"
,-,o,
~.-
,/\
/oT\
\ / \T/
/\
/t\
\ /\T/
. ..~. .
0
/ \
\o/
/ \
\/
0o-;%0
0
o°°r'~ ~ o
0
oo.o O 0
000
0
o o0~oo o ~
/ \ / \
\ /\/
/ \ / ",,,
222
2.8
Glueing $ la MOiler
(~)
Since we have seen a typical way of glueing together two components of Auslander-Reiten quivers, we want to exhibit some similar
processes of glueing, in this and in the next section.
In this sec-
tion, we try to give some insight into a construction which was used
very efficiently by
W. MHller
out paths of length
~ 2, he constructed a corresponding quasi-Frobe-
[25]. Namely, given any quiver with-
nius algebra with radical cube zero.
We will see that his method
amounts to a glueing of certain components.
Let
~,B,.--
A
be a finite quiver with vertices
without paths of length
ted cycles and
A
kA
A
Equivalently, A
has radical square zero.
is either a sink or a source.
lowing quiver
~ 2.
of type
r,s,..., and arrows
has no orien-
Note that any vertex of
For example, we may take the fol-
~8' and we outline also its preprojective
component:
o7\/o\/o\
6
2
7
8 f
3
4
~
9
Consider also the quiver
vertices by
A'
with reversed orientation:
r' ,s ',..., the arrows by
the preinjective component
~' ,B' , ....
denote its
Here we outline
223
]i
5'
V'
O'
2'
6'
~ 7'
3'
8'
4'
9'
We form now the quiver
r
a sink in
A
F = 4 u 4'
(and therefore a source in
following relations:
~
p~
= < r I ~,~'Is >.
same starting point, say
tion, for
r
r
see that
E(r)
M 4 U M4,
belonging to
Let
I
a sink in
°'" ~-mo~
with
E(r').
Thus we
MR, the only indecomposable not
Pr' for
r
a source.
On the other
M 4 n M4, are the simple modules
4 (and a source in
MR
Then by construc-
4, the indecomposable projective R-module
is cofinite in
Thus we see that
4
be the ideal gene-
R = kF/I.
is also injective, and has socle
M 4 U M4, being the
s
with same end
For every pair of arrows in
hand, the only indecomposables in
E(s), with
4
Or~
a source in
with head
for
rs
consider p
= <tj~,~'It'> - <tI~,~'It'>.
Nv
rated by all these relations p~B,p ~ and
P
r = r'
4'), and consider the
For every pair of arrows in
point, say
consider
with identification
A').
has the following Auslander-Reiten quiver
A - mo d.l""
224
where we have encircled the projective-injective modules.
the middle component has the following property:
Note that
if we omit the pro-
jective-injective modules, then the remaining quiver (it is called the
stable part) is of the form
Lemma:
Let
F
An, Dn, E6, E7, or
~F.
Thus we have seen:
be any connected quiver which is not of type
E8
there exists an algebra
and which has no paths of length
R
~ 2.
Then
and a component of its Auslander-Reiten
quiver whose stable part is of the form
~F.
This seems to be of interest with respect to the recent results
of Riedtmann
[32].
The glueing process outlined above is one-half of what MHller
actually did:
tive
he also glued the remaining two ends:
the preprojec-
A'-modules and the preinjective A-modules along their respective
simple modules, using again additional projective-injective modules.
In the example below, we obtain the quiver
6
2
7
8 ~f' E
3
4
9
with relations:
~'
= BB',
~'
= ge',
~'B = y'y =6'6,
= o = ~'~
for
~N'
e'c = ~'~,
=
~';
~'~ = v'~;
~ # ~ C {~,B,y,6,c,~,v}
An extension of this method to arbitrary quivers
oriented cycles has been given recently by Tachikawa
A
without
(see these p r o -
225
ceedings
[38]).
2.9
Glueing using splitting zero relations
We come back to the consideration of splitting zero relations
and want to show the behaviour of the corresponding Auslander-Reiten
quiver.
We only consider the case of a connected quiver
splitting zero relation
tions of
(F,p)
0.
As we have seen in
can be derived from some quiver
by identifying a subquiver of
(F,p)
g
is of tame representation
(i)
A
is connected,
An , Dn ,
(ii)
A
E6,
E7
decomposes
F
with one
1.3, the representaA
without relation
either of type A
or D . In case
n
n
type, there are three cases possible:
and therefore tame, thus of type
or
~8"
into
a tame quiver
Al
and
a quiver
A2
of finite type.
(iii)
A
decomposes
into two tame quivers
Al
and
A2"
The last two cases deal with situations we are already familiar
with, namely
whereas
(iii)
(ii)
provides a finite enlargement of the quiver
is the case of a glueing of the quivers
Al
A]'
and A2"
However, here the glueing is established without the introduction of
additional modules.
Let us just give two examples:
First, consider the following quiver
o~1
/ "
O(
C~2
C~3
c%
(with relation, as indicated, ~]~2~3~4e5~6
= 0).
~
<
"
i~
o
o~6
@'~--'-0
AI
O(
c~5
O~
joint union of the quiver
O<
of type
I<
~8
Here,A
is the dis-
226
and the quiver
42
of type
~7
o
We have to identity the full subcategories
MkA 2
of
which are of the form
type
MkA3, with
A3
U
of
MkA | and
V
of
being the following quiver
D6 :
o+--~--o+--o+--o
Note, that
U
lies in the preinjective component of
MkA I
. . . . .
whereas
Thus
MkF
V
lies
in the preprojective
MkA2
has the following component which is obtained from the
preprojective component of
Mk~
component of
MkA | and the preinjective component of
by the identification of the subcategories
>oji
Similarly, the quiver
P
U
and
V.
227
has subquivers
A l of type
in the subquiver
A3
~8' and
of type
Al
MkA 3
MkA 2
of type
and
A3
o
MkAl
in order to form
which intersect
~7
A5
A2
o
and
A2
o
are glued along the joint full subcategory
MkF.
The glued component of
MkF
has the
following form:
In the first case
is obtained from that of
jective component of
MkA
itself.
MkA
(i), the Auslander-Reiten quiver of
MkA
correspondingly:
MkF
this time, the prein-
is glued to the preprojective component of
For example, let
F
be given by
228
with
0
]
31
4
~i~2~3~4 = o
5
2
then
F
A
is a quiver of type
is as follows:
~9' and the Auslander-Reiten
quiver of
229
3.
REGULAR ENLARGEI~NTS
We are going to study the subspace categories
where
F
is a tame connected quiver and
MF
U(Hom(MF,MF))
is regular. The pattern
which occur in this situation seem to be of independent interest. We
will encounter a large class of non-domestic tame algebras, which, however, fall into a small number of similarity classes. These similarity
classes will be labelled using the extended Dynkin diagrams.
Before we recall some properties of the regular representations
of the tame quivers, let us introduce the notion of a pattern.
3.1
Pattern
We want to define an equivalence relation on the class of all
vectorspace categories, the equivalence classes being called pattern.
By definition, two vectorspaee categories
K
and
L
belong to the
same pattern provided there are full cofinite embeddings
L --+ K
of vectorspace categories
K--+ L
(note that an embedding
and
~ : K--+ K
of vectorspace categories is assumed to be compatible with the given
forget functors to the category of vectorspaces, thus there should be
a canonical isomorphism
pattern of
If(X) I ~ IXI
K will be denoted by
for any object
X
in
K). The
[K].
Let us make the following remark: in considering pattern, we only
will be interested in vectorspace categories
for which there exists a cofinite embedding
categories such that
n ~0m(K)
K
which are infinite and
~ : K--+ K
of vectorspace
is finite.
m
Note that if two vectorspace categories
K
and
L
belong to the
U(K)
same pattern, then the corresponding subspace categories
U(L)
and
can be derived from each other: namely, there are full embeddings
U(K) --+ U(L)
and
U(L) --+ U(K)
which can be controlled rather easily.
There is an additional situation where two vectorspace categories
K
and
L
U(K)
yield rather similar subspace categories
and
U(L).
Namely, recall that we have constructed rather frequently categorical
equivalences of the form
dimensional algebras with
M R ~ M(kMs), where
S
R
and
domestic (usually, S
S
are finite
will be the path
algebra of a tame quiver). We know that there exists a representation
equivalence between a full subcategory
V
of
M(kMs)
and
230
U(Hom(Ms,Ms)), where the complement of
V
in
M(kM S)
certain S-modules (those in the right annihilator
case, we will say that
MR
can be reduced to
there are vectorspace categories
U(K), and
reduced to
pattern
[K]
and
M
[L]
For example, let
RoP
K
and
L
consists of
r(Ms) .)
In this
U(Hom(Ms,Ms)). In case
such that
MR
can be
U(L), we will call the
can be reduced to
similar.
R
be the one-relation algebra
/%
Let
T
MS
S
be the path algebra of the quiver obtained by deleting
the path algebra obtained from the dual quiver by deleting
and
NT
be the corresponding modules with
a, and
b. Let
M R ~ M(kMs),
MRO p ~ M(kNT). Thus
S:
T:
>
o ,
~
o---~o---+o---+o+---o+---~+---~ ,
By definition, the pattern
1
dim MS= 1 / 1 - - 1 - - 1 ~ . 1
,
dim NT= o - - - t - - l - - l - - l - - t - - - o
[Hom(Ms,Ms)]
and
[Hom(NT,MT)]
.
are sim-
ilar. In the next sections, we will calculate vectorspace categories
of the form
Hom(Ms,M S)
rather explicitly, and we will see that sim-
ilar pattern may appear to look rather differently. However, we stress
the fact that for vectorspace categories
[K], ILl, the subspace categories
U(K)
K, i
and
with similar pattern
U(L)
are not too dif-
ferent.
3.2
THE REGULAR MODULES OF A TAME QUIVER
Let
F
N
be a tame connected quiver,
thus
F
is of type
An, Dn,
E 6 , E 7 , or ~8" Recall that a F-module has been called regular, provided it contains no indecomposable direct sun,hand which is preprojective
231
or preinjective. It follows directly from the definition that for
Y
iff
indecomposable, and
Y
f : X--+ Y
irreducible, then
X
X,
is regular
is regular. Namely, the set of indecomposable regular modules
is the union of full components of the Auslander-Reiten quiver.
Proposition.
category
R
Let
F
of regular
be a tame connected quiver. The full sub-
kF-modules is an abelian category, and is the
direct sum of indecomposable categories
and of global dimension I. The number
Rt
nt
all of which are serial
of simple objects in
is finite, and is different from 1 for at most three
Rt
t.
Recall that an abelian category is called serial provided any
indecomposable object has a unique composition series. In this case,
any indecomposable object is uniquely determined by its socle and its
composition length. If we consider a regular representation
then we call its socle
SOCRX
socle, and its length inside
inside the category
R
U
R
Since
R
the regular
MkF. Similarly, the
U
U(m)
the unique regular module
and regular length
m.
does not contain projective or injective modules, any
indecomposable regular module
X
has both an Auslander-Reiten sequence
ending with, and an Auslander-Reiten sequence starting with
R
F
will be called the simple regular modules. If
is simple regular, we denote by
with regular socle
of
the regular lens th~ in order to dis-
tinguish from the notions of socle and length in
simple objects in
R
X
X. Since
is closed under irreducible maps, these Auslander-Reiten sequences
lie inside
R, and therefore can be calculated inside
R: they are of
the form
o--+ U(m) --+ U(m+]) $ U(m)/sOCRU(m) --+ U(m+l)/SOCRU(m+l) --+ o.
Namely, it is easy to check that these sequences have the necessary
lifting property inside
R. Consequently, A U(m) = U(m+1)/SOCRU(m+])
and the irreducible maps are the inclusions
projections
belongs to
~im:~n~
R
:~:~)~~+I)~
~r
i
(
number with this property, thus we call
modules of the form
the orbit of
ASU(m)
U(m) ---+ U(m+l)
with
nt
o ~ s < nt
all
m ~ ]. If
and
nt
and the
U(m)
is the smallest
the period of
U(m). The
will be said to form
U(m).
The indecomposable modules in any
Rt
form a complete component
of the Auslander-Reiten quiver, and the form of this component only
232
depends on
n t. These are the first cases
identified
nt
n
t
(the dotted lines have to be
in order to form a cylinder):
l
=
>
>
3
o..
S
In the next section, we will consider vectorspace
the form
Hom(MF,MF)
quiver, and
MF
Note that if
is regular.
M
If
M = @ Mi
M
Let
is
F
with
r
categories
of
is a tame connected
Let us state some general assertions.
is an indecomposable
regular length of
Le~na I.
in great detail, where
~ nt
regular module in
if and only if
End(M) = k.
be a tame connected quiver, and
End(M i) = k, then
Rt, then the
Hom(MF,MF)
MF
regular.
is a Schurian vec-
torspace category.
Proof:
For
X
indecomposable
lated by the radical of
Hom(M,X)
anyway
End(X),
= 0 Hom(Mi,X ) . If
X
regular,
Hom(Mi,X )
is annihi-
thus the same is true for
is indecomposable
and not regular,
then
End(X) = k.
Lemma 2.
a reflection
Let
P
be a tame connected quiver,
functor. Let
space categories
MF
Hom(MF,Mr)
and
be a regular F-module.
and
Hom(oM,M F )
o : M F --+ M F
Then the vector-
belong to the same
233
pattern.
Proof:
F. Then
o
Let
o = Or, where we may assume that
Hom(MF,MF)
into
Hom(~M,M F)
Hom(~M,MoF),
Lemma 3.
Let
r
full embedding
Proof:
End(Mi) = k
Hom(Mr,Mp)
~ : K--+ K
If
x
Hom(oM,EoF(r)).
be a tame connected quiver. Let
M = ~ M.l wi~h
vectorspace category
by
with
for all
Mr
regular,
i. We denote the
K. Then there exists a cofinite
~n(K) =
n
nE
is simple regular, let
posable regular module with regular socle
p
is a sink in
and the only indecomposable object of
not being in the image, is
and assume
Let
r
defines a full embedding of the vectorspace category
0.
X(m)
X
be the indecom-
and regular length
m.
be the smallest common multiple of the periods of the simple
regular r-modules. We denote by
objects
Hom(MF,YF)
be given by
with
A p, thus
ally isomorphic to
K'
the full subcategory of
preinjective. On
K
K', we define
of all
~
to
~(Hom(MF,YF)) = Hom(MF,APY) , which is canonic-
Hom(APM,APY) ~ Hom(M,Y), since
simple regular with
Hom(M,X(m+1)).
YF
Hom(M,X(m)) # O, define
It is easy to see how
~
APM ~ M. For
X
~(Hom(M,X(m)) =
has to be defined on morphisms
in order to be functorial, and that it has the desired properties.
3.3
Calculations of pattern
It is easy to calculate the pattern of
ular F-module
The case (~7,3). We consider a quiver
simple regular P-module
possible
r
Hom(MF,M F)
for any reg-
M F. Let us show this in great detail in one example.
MF
F
of type
~7' and a
of period 3. Of course, there are many
(using the different orientations of
~7), and for every
£, there are three different simple regular modules of period 3. Since
all the different
MT
are obtained from any one of them by the use of
reflection functors, we may choose an arbitrary one. Thus, let us
choose the "subspace orientation", thus
F
is
6
5
o I
O
~O
2
3
7 0 " " - ' - ~ C,~-'--'-O <
4
0
7
234
and let
MF
be the simple regular module
O
>k---+k--+k*~---k< k+---o .
In order to calculate
F-module
socle
X
with
M, or else
Hom(MF,MF), we note that for an indecomposable
Hom(MF,X F) # O, either
X
X
is regular with regular
is preinjective. As before, let
the (unique) regular F-module with regular socle
n, and let
Hom(M,M(n))
~n : M(n) --+ M(n+])
M
M(n)
denote
and regular length
be the inclusion map. Since
is one-dimensional over
k, and
Hom(M,~ n)
is an iso-
morphism of vectorspaces, we see that the chain of inclusions
~l
~2
M(1) ----+ M(2) ----+ M(3)
gives raise under
Hom(M, --)
Next, we calculate
Hom(M,X)
X = AJl(r)
for some
j E ~
> ...
to the following vectorspace category
for
X
and some
indecomposable preinjective. Now,
0 < r < 7. Since we know that
Hom(M,X) = Hom(M,AJI(r)) ~ Hom(A-JM,l(r)) ~ (A-JM) *
r
and
A3M ~ M, we only have to determine the following dimension types:
0
dim
M
=
( 0
1 1 ] I 1 0 )
d i m A-1M =
1
( 1 ] I 2 I I l )
dim A-2M =
l
( o o I I 1 o o )
As we know, the preinjective component of the Auslander-Reiten quiver
has the following shape:
We obtain those indecomposable objects
Hom(M,X)
in our vectorspace
235
category
Hom(M,M)
which come from preinjective
leting the encircled points
F-modules
(for these modules Y, we have
X, by deHom(M,Y)=O),
and all the remaining points with the exception of the points
squares become one-dimensional
in
Hom(M,M),
whereas
in
the points in
squares become two-dimensional.
The fact that
M
very regular.
the maps
has period 3 implies that the obtained pattern is
Of course, we will use this fact in order to determine
Hom(M,f)
where
f : X --+ Y
is a map between indecomposable
preinjective
r-modules.
In order to do so, we consider
category
U
of all modules
Since in
U
any map is a sum of compositions
AJl(r)
with
we may again work with the corresponding
the full sub-
0 < r < 7, and
0 < j <_ 3.
of irreducible maps,
part of the Auslander-Reiten
quiver :
o
f
\
/'
o,.°ooo
/
122.332.1
1
11222~I
t
0111 i 11
I IfO000
"1~33221
-4~.~Zll
-t ! ! ! II0
ooooi~
o
[--- o ' ~
236
It turns out that for any indecomposable
XF,Y F
in
U
with
Hom(MF,X F) # O,
Hom(M~,YF) # O, and any irreducible map
the induced map
Hom(MF,f) # O. The maps in
Hom(M,U)
f : ~
~
YF'
are therefore
generated by maps corresponding to the edges in
\
/
.......
~
",
. . . . . . . .
s S
x
//
With dotted lines, we have indicated the continuation to the left
given by the Auslander-Reiten translation. In a vectorspace category,
the composition of two non-zero maps between one-dimensional objects
has to be non-zero, so we only have to worry with respect to maps between objects where at least one is not one~imensional.
Here we can
use now the relations given by the Auslander-Reiten sequences: they
immediately yield
3
3
~i~i = O,
B2y 2 = O,
i=I
~ Yi6i = O .
i=l
Also we have in addition
B I y 1 = O,
Namely, let
[31 = Hom(M,f),
and
B3y 3 = O.
3,I = Hom(M,g), where
irreducible maps
1
o o 1 1 1 1 1
1 1 2 2 2 2 1
o 111111
1 2 2 2 2 2 1
f, f', g, g'
are
237
with
fg + f'g' = O. Thus $i~f] = Hom(M,fg) = --Hom(M,f'g') factors
1
through Hom(M, o o i 1 ] ] I ) = O. And similarly, for B3y 3.
Finally,
it remains to consider
Hom(M,M(n))
and any
Hom(M,AJI(r)).
of linear transformations.
F-homomorphism
the set of maps between any
We claim that this is the full set
This follows easily from the fact that any
M --+ AJI(r)
can be lifted to
M
M(n)
) AJI(r)
M(n)
(apply first
that
I(r)
A -j
to the inclusion
Me---> M(n)
and use the fact
is inj ective).
Remark:
Let us give some geometrical
result of our calculations
interpretation
above. Any F-module
to the
V =
V 2 --+ V 3 --+ V 4 -÷ V ° +-- V 7 +-- V 6 +-- V 5
for which all maps are injective,
V°
and
with seven prescribed
V 5 ~ V 6 ~ Vy,
may be considered
subspaces
V],...,V 7
as a vectorspace
such that
V2 c V3 ~ V4
thus as an S-space for the partially ordered set
(] ,3,3).
It is easy to see that the elements of
the elements
in the intersection
implies that for indecomposable
dim V 3 n V 6 ~ 2,
decomposable
Hom(MF,V F)
just correspond
to
V 3 N V 6. What we have shown above
V, we always will have
and that there is just one infinite sequence of in-
S-spaces
V = (Vo,V],...,Vy)
with
dim V 3 n V 6 = 2.
Similar results can be obtained by the choice of any other regular module of a tame quiver. For example,
assertion that for an indecomposable
intersection
(] J i,j !4)
V i n Vj
there is the well-known
quadrupel
for any two different
is at most one-dimensional.
"four subspace quiver"
o
o
A
(Vo,VI,...,V4) , the
subspaces
Vi, Vj
Here, we have to consider the
238
and the simple regular A-module
o
The i n v e s t i g a t i o n
is
the additive
of
ttom(M~,M A)
shows t h a t
category of a partially
dim Hom(MA,V/i) <. t
ordered
f o r any i n d e c o m p o s a b l e
General rules:
this
veetorspace
set.
category
That is,
VA.
The previous example is rather typical for the
actual work one has to do in calculating the vectorspace categories
Hom(MF,MF) , where
happens that
MF
is a regular module.
Hom(MF,MF)
In many cases it actually
is the additive category of a partially
ordered set. In this case, the procedure is even easier: As above, one
first determines
dim Hom(MF,XF)
for the indecomposable
F-modules, and
here one finds out that one deals with a partially ordered set. Thus,
it remains to check whether for
Hom(MF,X F) # O,
XF,Y F
indecomposable with
Hom(MF,Y F) # O, there is some
f : XF --+ YF
With
Hom(MF,f) # O. Note that there cannot be any additional relations.
XF,Y F
maps
map
both are preinjeetive,
If
it suffices to consider the irreducible
f. Note however that it may happen that there is an irreducible
f : XF --+ YF
Hom(MF,f)
of type
where
Hom(MF,X F) # 0 # Hom(MF,YF),
= O. For, consider the example of the quiver
~21'
and t h e r e g u l a r
k
module
such that
A
MA
,k
id
of period
MA
2. The p r e i n j e c t i v e
component of the A u s l a n d e r - R e i t e n
...
332
...
> 221
322
whereas, however, we obtain under
set
quiver
is of the form
> ll0
> 211
Hom(M&, -- )
~ lO0
the partially ordered
239
)
-
(For example,
M A --+ X A
let
-
~
o
-
)
,,,,
•
-
-
•
+
)
k
•
)
\
•
,
>
•
\
•
lies inside the unique submodule of
XA
of dimension type
X A --+ YA).
One other calculation has to be done rather carefully.
seen above that in the case of
preinjeetive,
in
Hom(Mr,M£)
MF
of type
(~7,3), and
one obtained as set of maps
this is only true in case
MF
We have
XF
regular,
Hom(Mr,X F) --+ Hom(Mp,Y F)
the full set of all linear transformations.
simple regular,
•
dim X = (21]), dim Y = (I]O). The image of any map
(101). But this is the kernel of the irreducible map
Y£
)
is simple regular.
In case
the situation may be more complicated
However,
Mr
is not
as we will see in
examples of the next sections.
Finally,
let us show the structure of two other classes of
pattern of the form
Hom(MT,MF),
with
M£
regular, which we will need
in the later discussion.
The case
quiver of type
consider
and
(Dn,2).
This should indicate that we deal with a
Dn, and a simple regular module of period 2. Let us
the quiver
dim M =
0I 1 ! ...
]
; I O . The only interesting
pattern is that coming from the preinjective
n = 8, the preinjective
component
part of the
component.
In case of
looks as follows
o
...
1
oooooo,
,,,,/Z,,,,/-,,,,/Zx/x,
~I~
'"
Q,[]~
"
,,,,'Oo,,, ',
I t ~ 1''',
240
and for the encircled modules
X, we have
Hom(M,X) = O. For all the
other indecomposable preinjective modules
Hom(M,Y)
Y, the vectorspace
is one-dimensional, and no additional irreducibel maps are
cancelled, thus we obtain the following partially ordered set
Note that this is precisely the partially ordered set obtained from the
quiver
A7
~ A7
is the
by adding all possible commutativity relations, where
A7-quiver
o---+o---+o >o---+o---+o---+o .
In general, denote by
A
the quiver
~
-..
o+---o of
n
type
An, with all arrows going in one direction. The preinjective
part of the vectorspace category
Hom(MF,MF), where
regular representation of period 2, and
in the same pattern as
F
~-An_ |. The cases
MF
of type
is a simple
~n' will always be
n = 5,6,7,8
are depicted
in 3.5.
The case
M = (Mi,~)
(~pq,|).
Let
F
be any quiver of type
~n' and
a simple regular module of period ]. Then for all indecom-
posable preinjeetive modules
X, the vectorspace
Hom(M,X)
is one-
dimensional. Namely, all components
AM ~ M. Also, if
X,Y
ist~an irreducible map
homomorphism from
necessarily
M. are one-dimensional, and
i
are indecomposable preinjective, and there exf : X-+
Hom(M,X)
to
Y,
then there also is a non-zero
Hom(M,Y)
in
Hom(Mr,MF)
simple injective, using reflection functors. Then
source of
(but not
Hom(M,f) # 0). For the proof, we may assume that
F, and
X = l(s)
Y
is
Y = l(r) for a
for some other vertex with an arrow
o
>o which gives rise to the irreducible map f : l(s) --+ l(r).
r
s
N
Now, if F is not of type Alq, then ~ # 0 for all regular modules
of period I, thus
Alq
and
~
=
(rib I ..... BqlS)
0
Hom(M,f) # O. On the other hand, if
for our
M, then
~B ...~B
is the other path from q r
gives rise to another non-zero homomorphism
Hom(M,g) # O.
# 0
t~
s
in
in
F
is of type
M, where
F. This path
g : l(s) --+ l(r), and
241
It follows that the preinjective part of the vectorspace category
Hom(MF,MF)
ist just the additive category of the partially ordered set
obtained from
~ F
by adding all possible c o ~ u t a t i v i t y relations.
In particular, the vectorspace category
Hom(Mr,M F)
F, and not on the isomorphism class of
M
only depends on
(note that there is a one-
parameter family of such modules). Examples of these pattern will be
N
found in 3.5. This is the case of
lines in order to f o r
3.4.
a cylinder):
Some n o n - d o m e s t i c
We w i l I
F = A34 (with identified dotted
encounter,
patter n
in the sequel,
many p a t t e r n
which are
tame,
but non-domestic.
In this
section,
we w a n t t o e x h i b i t
it
turn
that
will
out
any other
a s m a l l number o f s u c h p a t t e r n ,
w h i c h we l a t e r
have to consider
is
and
sim-
ilar to one of those of this section.
The vectorspace categories discussed here are not additive categories of partially ordered sets, so it is difficult to decide whether
they are tame or not. Actually, for two of them (the cases
~n)~
this is k n o w
~
and
q
for a long time [29], also that they are non-
domestic. For the remaining ones we have to postpone the proof of the
tameness to the next section, where we will see that their pattern are
similar to pattern of tame partially ordered sets, thus also tame.
Actually, all other tame vectorspace categories (but one) which we will
have to consider will be additive categories of partially ordered sets.
Here, we concentrate on the point that for the non-domestic
vectorspace categories
K
which we will encounter, no domestication
will be possible. All have the property that there are either infinitely many pairwise different 2-dimensional objects with endomorphism
ring
k, or that there are infinitely many pairwise different full
242
embeddings of the additive category of one of the partially ordered
sets
(1,1,1,1),
(2,2,2), (1,3,3), (1,2,5) or (N,4)° Of course, these
embeddings give rise to infinitely many series which immediately shows
that an algebra
R
U(K)
which reduces to
cannot be domestic. How-
ever, we will see that there are finite full subcategories
K.
of
K
I
(and, again, infinitely many) such that any
K.
itself is non-domes-
i
tic, and in fact that the whole of
U(K)
can be rebuilt inside
U(K i) : there is a cofinite subcategory of
tion equivalent to a subcategory of
U(K)
We consider the following cases: always, S
algebra of a tame connected quiver, and
in two c a s e s ( ~
and~)the
which is representa-
U(Ki).
MS
will be the path
a regular module which is
direct sum of two simple modules, in all
other cases indecomposable. Besides writing down
S
and
MS, for the
benefit of the reader, we also note the quiver with relations with
algebra
R
such that
case
MR
S
reduces to
M(kMs).
dim MS
R
....
q
°1o...o
ac~'
On ~ 0 o
>0...0
....
>0 > ~
=
88'
= 0 =
8B'
=
0
000...0111
o
1
ololo,
o
~ i'~:~...o
~a'
243
o
o
66
~ o + . ~ o
c
o
olllo
c~c(' = B6'
T
Q
o-----~o---~o<
~8
o----~
~
o<
c
o
o
11111]o
1
>o---*o
~o+---o+--~<
= ¥Y'
o
ooi ] l 1 1 1
The calculation of the vectorspace category
~t
o\~
K = Hom(Ms,M s)
gives the following:
eq/-/Z.III
@n
..........~...y...~f>...y...~¢~...%
244
~7
Note that all of them contain countably many two-dimensional objects with endomorphism ring
k, and all but the cases ~nn and ~ 6
contain a countable number of subsets of the form
other hand,~6, ~
(2~2,2), and ~ 7
and~
7
andS8
such of the form
has subsets of the form
(1,1,1,1). On the
contain countably many subsets of the form
(1,3,3). Finally, ~ 8
also
(1,2,5) and (N,4). Of course, in this way we
obtain many one-parameter families of indecomposable objects in
U(Hom(Ms,Ms)).
We use now the vertex marked
to obtain an equivalence between
tions of a different bimodule
t
in the quiver of
MR
and the category of representa-
kXT . Thus, T
is obtained from the given
quiver with relation by deletion of the vertex
always a source), and
corresponding to
t
XT
t
(note that
t
XT
itself is always projective, and
even indecomposable). It turns out that
is of finite representation type (we will indicate
its complete Auslander-Reiten quiver), so that the subcategory of
which is canonically representation equivalent to
cofinite in
is
is the radical of the projective R-module
(note that
in the cases ~ n ' ~ 6 ' ~ 7 ' ~ 8
in all cases, T
R, in order
U(Hom(XT,MT)),
MR
is
MR .
Let us exhibit the Auslander-Reiten quiver of
the corresponding vectorspace category
MT, and indicate
L = Hom(XT,MT).
o~o
:
~
.
:(&N'ZX) m°H
>~o
\/
' ~ 1 V~ I,!I,,\!S
l!
"o-~..,
°~"
A ~o
St78
246
<>7>11
~
/ "<Z
)
)
II
b-t
X
(
I
k
~-.27
~eq~
eq
®
°i
'.-t
i
247
dim XT= o
MT
00o°
"100~
t•og
t 00~
/ \
1 I°o°
000~
Hom(XT,MT)
1
1
111 t
oo
m
°
'
~+rl' - ~
o ~"~ ' ~
a
°°
+i
cta'= B~' = yy'
"
o+
++
Ol
dim X T = 01!
Ol
oo
+
+
++
.
Yll
12
oo+
""e/
i~
z
~
+
2
o~
I~
i~
+
~c~
~
~z
o~
~z
~
oo
°/2\.;,,\+/\ /I°'\('\/ \ / \ / ,.\ / \ o, .+
oo
T:
+~++l
Hom (XT, ~T )
,~
]ase ~ 6
249
/ i
,-; ~ S k
-
I ,I°
0 '~
/
~o
~
.i°;
/
,,
J
0
"~-'0
O.v,
.~-,0
"-,. o
~t
\
II
!
,0
/
-~J.~
~
~ . ......., L : - /
~.
®
o
250
o
-tO 0
~0
~~
~"
II
I
/Z"o
o /
"-.
"~
/ 7
/
®
o
251
In all cases, we see that
full vectorspace
L = Hom(XT,MT)
subcategory into
K = Hom(Ms,Ms)
ber of ways,
with pairwise disjoint
us formulate
this result as a proposition.
type * (* = one o f - - - ~ D q ' ~ n ' ~
can be embedded as a
in a countable num-
images. For later references,
Note that in case
K
let
is of
.... ), then we call the pattern of
K
the pattern of type ~.
PROPOSITION:
For the pattern of type --~
- - ~ q ' ~ ' - -..
~ '-~-6 ' ~_7_' ~ '
there exists a vectorspace
vectorspace
category
disjoint vectorspace
L
category
K
in the pattern~
which is equivalent
subcategories
K.
of
i
subcategory
of
subcategory
of
Namely,
U(K)
U(L).
let
r(Ms),
and
W
equivalent
MR
and representation
in
V
and codomestic
cofinite
r(XT).
to
equivalent
in
MR
V
with no non-zero direct
Then
V
U(K), and
to U(L).
is codomestic
W
MR
in
is even cofinite
Thus
W, is representation
U(K)
subcategory of
to a (codomestic)
the full subcategory of objects of
no non-zero direct summand in
and representation
equivalent
L = Hom(XT,MT) , as above. Let
be the full subcategory of objects of
summand in
to countably many pairwise
K, such that a cofinite
--
is representation
K = Hom(Ms,Ms) ,
and a finite
with
MR
in
V N W, being cofinite
equivalent both to a
and to a codomestic
subcategory of
U(L).
Note that the proposition
vectorspace
mestic.
categories
In fact,
construct
above implies,
in particular,
of type ~ - - ~ D q ' ~ n ' ~ n ' ~ 6 ' Q ' ~ 8
it allows for the finite subcategories
infinitely many pairwise different
that the
cannot be doKi
of
K
to
series.
Perhaps we should stress the following
implication
of the proposition:
nearly everything nice or pathological
what happens in
U(K)
happens
not only ones but at least a countable number of times: just consider
the various
uations
U(Ki). This is the reason for our feeling that these sit-
should be called non-domestic!
3.5
Regular modules
Let
F
F-modules
Hom(Mr,M r)
Mr
with
Hom(MF,MF)
tame.
be a tame connected quiver. We want to list all regular
Mr
such that the corresponding
is of tame representation
notation for the type of
vectorspace
category
type. We will use the following
M F. Assume that
F
is of type *
(one of
252
pq, Dn, E6, E 7, E8). If
say that
MF
Hom(MF,MF)
M
is simple regular, and of period
is of type (*,p). Note that for
will only depend on
*
and
p > ]
p. This, in fact, is an easy
consequence of Lemma 2 in 3.2, since for two quivers
the same type *, and
M
ular with same period
and
F'
a F-module, M'
F
M'
is of type (*,~). In case of
for the type of the module
MF
M = M' @ M"
type
~
with
M', M"
(Dn,(n-2) @ (n-2))
and a F-module
F
M
with
M', M"
(~pq,p @ p)
simple regular
(~pq,p , q)
will be
simple regular in different
will stand for
M = M' @ M"
is indecom-
p, then we say
F = ~pq, we write
M = M' @ M"
in the same orbit, but non-isomorphic, whereas
orbits. Similarly,
of
o. Also, we have seen in 3.3 that
posable regular, of regular length 2, and of period
the type of
F'
can be obtained from
there is only one pattern of type (~pq,|). In case
MF
and
a F'-module, both simple reg-
p, we can identify the underlying graphs of
in such a way that the module
by a sequence of reflection functors
that
p, we
the pattern of
with
M', M"
F
to be of
both simple reg-
n
ular in the same orbit, but non-isomorphic.
Also, in many cases there exists a Dynkin-diagram
A-module
NA, such that the
vectorspace category
F 0 A-module
Hom(M @ N,MF6A)
M @ N
In these cases, we also write down
&, note that
M'
(F 0 A, M $ N)
its type by
in case
MF
A
is essentially
always is indecomposable and
A, all components = k. The pair
will be called a completio ~ of
(*,p)
and a
which moreover is non-domestic.
unique. The corresponding A-module
has, for the noted orientation of
A
again gives a tame
is of type
(F,M), and we denote
(*,p). In case a comple-
tion exists, we write down the similarity class of the pattern of
(,,p), it is represented by the pattern of one of the vectorspace
categories introduced in the last section, and we call it the similarity type of
(*,p).
THEOREM 3. Le__t F
ular representation. If
be a tame connected quiver, and
Hom(MF,Mp)
is not wild then
MF
a reg-
(F,M) is of one
of the following types, and conversely, in all these cases, with the
possible exception of
(~4,1), Hom(Mp,M F)
iS tame.
253
type
Simil.
type of
pattern of Hom(MF,MF)
(*,p)
(~pq, P) ;
;
(A22, 1) :
:
(723, I)
:
. . . . .
0--'->0--+0"--+0
;
(A24, I) :
(~25,1)
0--+0
;
N
(A26, I) ;
~2q, l)
q~7
(~33,1)
:
Q
:
Z
(A34, I) =
(A35, I)
:
(A36 , I) :
(A44,1)
~
,,/
",V___Nf
~/
~/
x=
254
(D4,1) %
.
(5 6 , 2 )
.........
.
.
.
.
.
('D'7 ' 2 )
.
.
.
@
.......
~ / ' v v v v
v v v
x.
(~'8,2)
.........
Q
(~6,2)
--
@
:
@
('E7,3)
(g7' 4)
@
.........
.~
Q
255
(e8,5)
.
.
.
.
.
.
~.J' ~
,..
_:::_/YY_/_///A/
L
( ~ p q , P@p) . . . . . . . . . .
;
.
.
.
.
.
~Pq
(h' (n-2)* (n-2))
'
'2,
For the conventions for illustrating vectorspace categories, we
refer to 2.4. One particular case has to be explained separately, namely
(~4,.I) . Here
~
indicates a vectorspace category
with 8 one-dimensional and 2 two-dimensional objects with trivial endomorphism ring, and non-zero maps
•
|-~. . ~ |
256
3
~ ~.$. = O,
i=! ~ i
~ | $ I + X ~ 2 ~ 2 + ~ = O. This pattern is obtained for the following simple
satisfying the relations
$iYi = O for all i, and
regular representation
ko
(11)~
M =
° k ~ k k
or
of not.
the ~4-quiver
If
of
F, P'
£ =
(with
% # O,])
. We do not know whether it is tame
~ o
are quivers, M
a representation of
F', then the vectorspace category
the product of the categories
Hom(M $ M',MFop,)
Hom(M,MF)
and
A
and to the P-module
M
the A-module
the non-domestic vectorspaee category
module
N
N
Hom(N,MA)
form
o
with
n
.........
clearly is
F
a D~kin
dia-
in order to obtain
the vectorspace cate-
has the following form: For the quiver
+o...~
of the chain
one
Hom(M @ N,MFoA). For the
with all components one-dimensional,
gory
M'
Hom(M',MF,). This remark
will be used in those cases where we had to add to
gram
F, and
A
of the
edges, we obtain the additive category
with
denoted by (n)); for the quiver
n
elements (which we also have
o - - + o . ._. ~_
of type
Dn, we obtain
the additive category of the partially ordered set
- - ~
......
with
the pattern for the completion
(~22,1)
2n-2
of
points. For example,
(~22,1)
is given by
This is a partially ordered set of width four.
We will call an algebra
quiver
F
R
a regular enlarsement of a tame
provided there exists a P-module
M
with
M R ~ M(kM T)
such
that for any connected component
b
gram,
MIP i
MIF i
F. of F which is not a Dynkin dial
is regular. Of course, we always can assume that all
are non-zero. In case all but one connected components of
Dynkin diagrams, and
precise conditions on
components of
F
F
M
is not of type
for
R
are tame, then
F
~4' then theorem 3 gives the
to be tame. In case at least two
R
are
is tame if and only if
F
has
257
precisely these two components
following forms
least one
wise
R
FI,F2, and any
(~pq,p), (~22,1)
Mi[F i
is of the form
and
Mi]F i
is of one of the
(~n,n-2). In this case, if at
(~pq,p), then
R
is domestic, other-
is non-domestic. All these assertions follow directly from
Theorem 3.
For the proof of Theorem 3, we have to show that the listed cases
are tame, and that these are the only ones. The proof that the remaining cases all are wild, will be given in the next section. In the
present section, we concentrate on the listed cases, show that they
are tame, and deal with the corresponding similarity classes. In fact,
we consider first the similarity classes, since it will turn out that
nearly all contain pattern of additive categories of partially ordered
sets, so that the tameness follows from the theorem of Nazarova.
Calculation of the similarity type_~s
Let
(*,p)
be a type for which a completion is claimed to exist.
We have to exhibit an algebra
type
(*,p), and
R
such that
MRO p ~ M(kM~,) , where the pattern of
is of the indicated type ~ , ~ , ~ ,
R
M R ~ M(kMF)
or~.
with
MF
of
Hom(M~,,MF,)
It is of interest that for
we always can take a one-relation algebra, thus we just give the
corresponding number in Theorem 1.
% 2 , I)
76
(~36' 1)
(~23 ,I )
82
(~26 ,I )
79
8]
(~33,l)
77
(~n,n-2)
(~5,2)
(~6,2)
78
18
24
20
(~8 ,2)
~6,3)
(~7,4)
27
60
23
80
(~44'I)
(~24 ,I )
Proof of tameness.
If the vectorspace category
Hom(MF,M F)
is
the additive category of a partially ordered set, then we just use the
theorem of Nazarova in order to verify that
Hom(MF,M F)
is tame. Note
that this is very easy to check in our cases due to the fact that the
pattern which occur are periodic. Now the similarity of
%3,1),
of
(~7,3)and
(~24,l),
of
(~8,2)and
(~6,2)
(~23,I),
and
and
258
finally of
n-2
~n,n_2 )
and
(~n,n-1), show that all these pattern are
tame.
Thus there only remain two cases, namely the types
(~pq,p ~ q)
N
and
(Dn,(n-2) ~ (n-2)), which we also have denoted just by
~p
and (D'), respectively. For these two cases, we do not know any
q
direct reduction to the case of a partially ordered set, using the
concept of similarity of pattern. However, fortunately, both pattern
are known, for a long time, to be tame. Namely, they give rise to matri~ problems solved by Nazarova and Rojter in
[26]. The case ~ q
later also has been solved, with a different technique, by Donovan
and Freislich in
tion
[13]. Namely, we consider the following representa-
M = M' @ M"
~
where
M'
~
,
is the simple regular representation for which the map cor-
responding to
tion with
for
~
is
k--+ o
M(kM F) = MR, with
with relations
~'
k--+ o,
and
M"
the simple regular representa-
being the map corresponding to
R
B. Then
being given by the quiver
= O = 66'. The indecomposable R-modules can easily
be described by "strings" and "bands".
Remark.
The proof above essentially finishes the investigation
of the tame one-relation algebras. It remains to consider two special
algebras, namely 31 and 73, which will be done in 3.9. These two algebras are not themselves regular enlargements of tame quivers, but
specialisations of such enlargements, however not of one-relation algebras.
Note that not all pattern do occur for one relation algebras:
the regular enlargements of a tame quiver
M
£
by a decomposable modules
will always be defined by more than one relation, thus the non-
domestic cases ~ q
and _.Q
algebras. Also, the case
are impossible for tame one-relation
~4,1)
will never lead to a one-relation
algebra. If we restrict to one-relation algebras with zero condition
259
or with strict commutativity condition, as in
also the pattern [E)
3.6.
[39]
or
[36], then
does not occur.
Proof of Theorem 3
It remains to show that the listed cases are the only ones which
can be tame. We will need the following lemma.
Lemma.
Let
X,Y
vectorspace category
= I, and
be non-isomorphic two-dimensional objects in a
K
with
o # f C Hom(X,Y)
Proof:
Let
O # f C Hom(X,Y), and
lying vectorspace
Y2 = f(x2)
End(X) = k = End(Y). If
is bijective, then
IxI
of
x. Since
is a basis of
Y. Since
as follows: If
(X @ Y)), where
U
generated by the elements
@(v) ~
(O,Y2), with
with the map
~
Hom(Y,X) = O. Define an embedding
matrices
F
be given by the object
v(~ (xl,Y2)
(V,~,~)
to
and
V ~ (X 0 Y)
v ~ ) (x2,0)+q)(v) ~ (O,y|)+
given by the matrix
(V',~',~')
of
Mk~, a homo-
i s g i v e n by a p a i r of
b2o]
b, O
la,,
a22
Ioo
0
with
and
F(V',~O',~,b')
(A,B), say
A
is a vectorspace with two
is the subspaee of
V 0 V --+ V @ V @ V @ V
F(V,~,~)
Yl = f(xl)'
the inclusion maps. Note that we can identify
and t h e n f o r two o b j e c t s
raorphism from
a basis of the under-
are not isomorphic, and
(V,~,@)
endomorphisms, let its image under
(U,~, V ~
Xl,X 2
dim Hom(X,Y) =
is wild.
is bijective,
X,Y
End(X) = k, it is easy to see that
F : Mk~ --+ U(K)
f
K
b3
O
b3J
aij,b i E End(V), satisfying
(00o) (o0o )
I qO
But this implies that
1 ~0' ~'
A.
all = a22 = b] = b3,
allq) = q)'a]], al|@' = @a]1 , thus
to
(V',q)',@'), and its image under
a|]
F
a12 = a21 = b 2 = O, and
is a map in
is just
Mk~
(A,B).
from
(V,q),@)
260
We start with considering one special case, namely we show that
(AI,I,I)
is wild. Thus, consider
F =
O
• and
M
I 0
(0 1)
k k @ k k
X 1
(0
The corresponding algebra
with relations
~'
R
= $6',
~,B
~,)
~i
reducing to
M(kM F) is given by
B~' = O, independently from
first obtain the relations
can replace
with
.
~'+%~B'-~B'
= O,
%~'-B~'
k. (For, we
= O, but we
by linear combinations in order to get the relations
above). We consider now representations of an
E7-quiver
U
f
U1 ¢---+ U2<<-- U3¢---~ V -->>U 4 +--DU 5 -->>U 6 +---°U 7
with maps being monomorphisms and epimorphisms as indicated, and define
a funetor into
MR
by sending this representation to
0
0
0
"00
0 1000
0.0
00.
O"
U1U3Uo. f
U2VVU5U7 /
~
~
Ci°i)
*
O0
0
0
"0 0 O r
1 O0
00.
000
In this way, we obtain an exact embedding into
MR
VU4U6
o)
which is a repre-
sentation equivalence with the full subcategory of images, thus
wild.
R
is
(The corresponding case of the local algebra
R = k <X,Y> / (X2+y 2, YX, y3)
has been treated in
[~];
the proof
that the functor defined above is a representation equivalence, can be
found in that paper).
As a consequence, if
F = Apq, and
MF
is indecomposable of
261
period
q
and regular length
may suppose that
Mr
q+l, then
is wild. For, we
Hom(MF,M F)
is of the form
id~kk
id> kk ... k k ~ d
kk"
--
(10)
> k
...
kk ,
(1)
but then the corresponding algebra is given by
~,
~2
~1/~o---+o ... O ~ p
~'
BI~
~o
g2
with relations
~p...al~' = ~q...BIB',
previous case by shrinking the arrows
~i ~' = 0
~i,Bi
We consider now the general case. Let
of tame type, and
that
Hom(MF,MF)
Mp = (Mi,~a)
with
F
i ~ 2.
be a connected quiver
a regular representation of
F, such
is not wild.
First, we claim that
M
cannot be the direct sum of two iso-
morphic simple regular modules. For, if
regular, let
specializes to the
N(2)
M = N • N
with
N
simple
be indecomposable regular with regular socle
N
and regular length 2, then we can apply the previous lemma to
X = Hom(M,N)
and
Y = Hom(M,N(2))
in the vectorspace category
Hom(Mp,~F), and get a contradiction.
Similarly,
M
cannot be the direct sum of three non-zero reg-
ular modules. For, assume
M = N @ N' 0 N", with
regular. By the result above,
isomorphic. Again, denote by
of regular length
Hom(MF,M F)
m
N, N', N"
N(m)
simple
have to be pairwise non-
the indecomposable regular module
with regular socle
N. Then we obtain in
as full subcategories partially ordered sets of the form
(n, n, n), namely, take just all
Hom(MF,N"(m)) , with
Hom(MF,N(m)),
Hom(MF,N'(m)),
m j n. But according to the theorem of Nazarova,
this is impossible. In general, M
then
N, N', N"
Hom(N 0 N' • N",M)
will map onto
N $ N' ~ N", and
can be considered as a subcategory of
Hom(M,M) .
Next, we note that
let
!(i)
dim M. < 2 for all vertices i. Namely,
I be the indecomposable injective representation correspond-
ing to the vertex
i. Then
dim Hom(M,l(i)) = dim Mi,
and the
262
endomorphism ring of
Hom(M,l(i))
as an object in
Hom(M,M F)
is
k.
Thus we can apply the lenmm in 2.4.
Also, we see that
M
cannot have two isomorphic regular composition
factors. Namely, by previous considerations, we know that for a decomposition
M = M' ~ M", the regular composition factors of
different from the regular composition factors of
Thus, assume
M
of
M
is
are pairwise
is indecomposable and has two isomorphic regular compo-
sition factors, say of the form
regular length
M'
M".
S
with period
t, such that
M
is of
~ t+l. Actually, we may suppose that the regular length
t+|. In case
F
is of type
~mq' we have
t = I, p, or q.
The last two cases are impossible, as we have seen above. Similarly, the
first case is impossible, since also in this case
M(kM F)
would specialize
]
to some
M(kNA), with
NA
of type
~l]'
i)" In cases
Dn, E6, E7, E8, one
easily observes that for an indecomposable regular module
and regular length
is of dimension
of
M
> 3.
M
is equal to
are pairwise non-isomorphic,
consisting of all objects
Hom(M,X)
M
M
is
< 2. Namely, assume the
3. Since the regular composition factors
the full subcategory of
with
X
Hom(MF,MF)
regular, is of the form
°..
depending whether
of period t
t+l, always at least one component of M
It follows that the regular length of
regular length of
M
or
...
has 3, 2, or l indecomposable sun,hands, respectively.
But in all cases, it is easy to find subsets of the form (n,n,n) for all
n£~.
263
If there exists an arrow
o
r
> o
s
in
F
with
dim M
r
= dim M
s
= 2~
N
and
s
~
: Mr
~ Ms
an isomorphism,
is a sink, and
M"
M = M' $ M", with
simple regular of period
F = ~lq
from
' with
then
r
a source,
r to s. Otherwise,
M'
q, and
and
s
the vertex
a sink,
. Namely,
Hom(X,Y)
can be at most one-dimensional,
is one-dimensional,
The assumption
~
r to s
and therefore
where
~,
also
X = Hom(M,l(s)),
to be bijective,
implies
that also
~*
By the lemma at the beginning of this section we can conclude
Hom(M,M)
d i m M' = I o! ... o! I ,
.°.
component of
M
easily observes
that for
are isomorphisms.
wi~h
period
M
is of type
Namely,
dim M r = dim M s = 2
o
~ o
of period 2.
cannot occur,
(~pq,p,q),
say
M = M' @ M"
1 I ... 1 1 , or else at most one
o .°. o
dim M"=
in all other cases one
w i t h different vertices
joining
This is clear for
indecomposable
n-2
M
may be two-dimensional.
there is a chain of arrows
~.
can be identified with an element
is wild.
F i n a l l y we note that either
with
~*
path
given by
in
that
but
I, and
only in case
the dual map
Hom(X,Y).
is bijective,
on
is a source,
there is an additional
there is just one path from
Hom(I(s),I(r))
r
simple regular of period
M' ~ M"
and thus
Y = Hom(M,I(r)).
F = Alq,
r and s
such that all
E 6, E 7, E 8, and also for
Note that the case
since we recall
that
M
length
~
r,s,
~
with
n
is of regular
~n'
M
of
length
at most 2.
We have shown that
regular composition
M
factors,
have at most two-dimensional
(~q,p,q)__
(A~pq,l)
the p a t t e r n
and
of
M
M
This finishes
AmM
is of type
is two-dimensional.
exclude all cases not listed in theorem 3, but the
(~n,2). For these two situations,
vectorspace
categories
we have calculated
p,q,n,
are wild, using the theorem of
and that they are tame precisely
3.7.
translates
and that either
in section 3.3. It follows that for large values of
the corresponding
Nazarova,
2, not two isomorphic
that all A u s l a n d e r - R e i t e n
components,
or else that at most one component
Now these conditions
cases
has regular
in the listed cases.
the proof.
Some components
In the preceding
of the A u s l a n d e r - R e i t e n
sections,
we have discussed
quiver
the question under
264
what conditions a regular enlargement of a tame quiver is tame, again.
In case
R
is even domestic, we want to describe the category of
R-
modules in more detail. In fact, we will describe certain types of components of the Auslander-Reiten quiver of regular enlargements of tame
quivers.
Thus, let
with
Mp
F
be a tame connected quiver, and
regular. If we consider
TMF
a bimodule
M(TMT), then we know from 2.6 that
nearly all components of the Auslander-Reiten quiver of
unchanged in the Auslander-Reiten quiver of
MF
remain
M(TMp). Namely, the only
ones which can, and will, be changed are the preinjective component of
Mp, and those regular components of
of
MF
which contain direct summands
Mp.
Let us consider first the case where
with
R
F
a tame connected quiver, and
T
MF
is simple regular,
being of type
Am, so that
is given by a quiver with relations of the following form
m
where
MR
m-I
"
reduces to
M(TMF). We describe now the component of the
Auslander-Reiten quiver of
M F. Denote by
Epm
~
R
containing
M. Let
p
be the period of
the following quiver
BP+2 I
As in 2.2, we form
~ ~
but we reverse now the orientation of all
pm'
the arrows of the form (Bi,z) and (~,z), with z E N. Denote the
265
new quiver by
quiver
p = 3, m = 3, we obtain the
(~ ~pm )'. For example, for
(~ Z33 ) '
where the two dotted lines have to be identified in order to form a
cylinder.
Len~na 1.
which contains
The component of the Auslander-Reiten quiver for
M, is of the form
R
ON ~pm)'.
For the proof, we consider first the ring
R'
obtained from
R
by reversing the orientation of all the arrows in the left arm, so that
MR,
is of the form
M(k(M @ N)poA), where
~
is the quiver
e >o ... o >o , and dim N A = (I 1 ... I ]). The vectorspace category
2
3
m-I m
-Hom(M • N,MF6~) is the disjoint union of Hom(M,M F) and a chain with
m-|
7
elements. Now denote by
R
the category of regular F-modules, by
the category of preinjeetive F-modules. Now, as we know,
is the additive category of a chain, and any non-zero map
with
I
preinjective, can be factored along this chain. This implies
that any indecomposable object
: U~
Hom(M,R)
M--+ I,
(M @ N) ---+ X, either
in case
xIP
is regular, then
(U,~,X)
XIF
in
M(k(M $ N)pOA)
where
is preinjective or regular. Also,
XIF
is indecomposable, and
U
is one-
dimensional. From this it follows easily that the indecomposable objects
(U,~,X)
late this from
with
R'
XIF
to
regular, form one component. If we trans-
R (using reflection functors), we similarly
see that the indecomposable R-modules
ponent
as
Y
with
C
of
MF
quiver of
MR
and that those not belonging to
Mi(z) =(M(z) $ P(i))/ M
Y[P
in the same com-
Mp, form one component of the Auslander-Reiten
with
C
are of the form
z 6 ~, | ! i ! m, where
M(z)
is the
266
indecomposable regular module with regular length
M, and
P(i)
z
and regular socle
is the indecomposable projective module corresponding to
i. There are obvious maps between the modules in
C
and the
Mi(z),
so that we obtain a configuration of modules and maps of the form
(~ Epm)', and it is easy to see that any other map between these modules is a composition of the given maps. This then implies that we
really have constructed in this way the component completely, as we
wanted to do.
For example, for
p = 3, m = 3, denote
V, and use the notation
M(z), and
AM
by
U
and
Mi(z) =(M(z) ~ P(i~/M
A2M
by
as above.
Then the corresponding component is as follows:
P3 . . . . . . . . . .
P,~ (2) . . . . . . . . .
2~
~
c'~f~
~'~
P1 (3) . . . . . . . .
1
"
¢./~7~
~-
~ ' ~ M(2)
~
M ~
U(4)
U(3)
...
U(2)
u~
V(4)
V(3)
~
P3 ( 4 < ~
V(2)~
V a'~"'~"~
_
P3 (3) ~
P3(2)
P2 (4 < ' J
/
~
P2(3)
P3 . . . . . . . . .
P3(5)
~<~
(2) - ~ . . . . . . .
..°
P1 (4)
P1 (3) . . . . . . .
Such a component exists for
O-----+~_
>
-O )O
here, F
is an
>O O
>
~6-quiver, and the dimension types of
o
0
1
0
dim M = 000
111 ,
11
dim U = OOO
Similarly, we have such a component for
are
1
1
]OO ,
Ot
M, U, V
dim V = OOO
]10 .
OO
267
N
with
F
being of type
0
dim
A3q , and with
0
1
M = 001
1 ,
dim
0
U = 000
1...1
0
0
m
dim
V = 000
0...0
Note that the components of the form
ous: they contain
,
1
0
.
0...0
(~ Epm)'
are rather curi-
indecomposable projective modules, and every
other indecomposable module belonging to the component is obtained from
-i
them by applying some A . In this respect, they are similar to the
components of preprojective modules of quivers without relations. On
the other hand, whereas for an indecomposable preprojective module
we always have
P
End(P) = k, nearly all indecomposable objects in such
a component have non-trivial nilpotent endomorphisms. However, we note
that at least the following is true: any nilpotent endomorphism, or,
more general, any non-invertible homomorphism between two indecomposable objects in the component, is a composition of irreducible maps.
In the situation above, we have seen that for the indecomposable
objects
(U,~,X)
in
M(k(M @ N)FOA) , always
X[F
is either regular
or preinjeetive. This is a rather strong assertion, as the following
lermna shows.
Lemma 2.
let
MF
Let
r
be a tame connected quiver, A
be non-zero regular, and
posable object
(U,~,X)
in
Hom(NA,MA)
Proof.
with
I
MT
is
is simple regular
M
M 1 . Let
is not simple regular, say with a simple reg~] : M I --+ I
be a non-zero homomorphism,
indecomposable injective, and extend it to a homomorphism
: M--+ I. Let
object
if and only if
XIF
is the additive category of a chain.
Assume
ular submodule
arbitrary. Then for any indecom-
M(k(M 0 N)FOA), the restriction
either regular or preinjective,
and
NA
some quiver,
~ : M--+ M/M 1
(k, (~), I @ M/N I)
indecomposable, but
Also, if
in
I @ M/N I
Hom(NA,MA)
be the projection, and consider the
M(kMF). It is easy to see that it is
is neither preinjective nor regular.
is not the additive category of a chain,
then it either contains one two-dimensional object
Y
with endomorphlsm
268
ring of dimension
YI,Y2. In
! 2, or two one-dimensional
incomparable objects
Hom(MF,MF), we always have as non-zero objects
Hom(Mr,M F) =: X, and
Hom(Mr,IF)
posable injective module.
exists a subspace
respectively,
U
=: X', where
It is well-known
of the object
t, such that the tripel
(U,t,X @ X' ~ YI @ Y2 )
is some indecom-
X @ X' @ Y, or
of the vectorspace category
clusion map
Ir
(and easy to see) that there
X $ X' @ YI @ Y2'
Hom(M 8 N,MFOA), with in-
(U,t,X 8 X' 8 Y)
or
is indecomposable.
Let us consider now some cases of regular enlargements
M(k(M 0 N)FOA)
X
where there exist indecomposable objects
(U,~,X)
with
being the direct sum of a non-zero regular and a non-zero preinjec-
rive module. We only will consider domestic regular enlargements
quiver of the form
of a
Apq, but we hope that these examples will shed some
light on the general situation.
The examples will cover at least all
cases of domestic regular enlargements which occur in Theorem 1, thus,
in this way, we finish our program of giving a complete description of
the module categories of the domestic one-relation algebras occurring
in Theorem I.
We start with the following one-relation algebra
R :
.....a..
> ~
~
= O.
V
Here, £
is the
(I ]), and
&
the vertices
and
MpO g)
~ll-quiver
~
with
MF
the disjoint union of the two
u, v, with
U,V
consisting of
simple modules~
M R ~ M(k(M @ U @ V)FOA). The vectorspace category
Hom(M @ U @ V,
is of the form
•
,
•
.,o
,i,
where the long chain is formed by the objects
Hom(M,Ii) , with
P
Al-quivers
being the corresponding
m ......
of
being of dimension type
Mi
Hom(M,Mi)
and
being the indecomposable regular representation
with regular socle
M
and regular length
i, and
I.
the in-
i
decomposable
representation of
F
of dimension type
(i i-I). This
shows that we can construct indecomposable R-modules as follows: Let
XI,X 2
be non-isomorphic
indecomposable
F O A-modules with
269
Hom(M 0 U 0 V,X i)
note by
one-dimensional, say generated by
X'l the R-module
Hom(M @ U • V,X2)
(k,~i,Xi). If
otherwise let
(:(:,
X|X 2
XIX 2
and
denote the R-module
be the R-module
X1 ~ X2 0 U • V), w i t h
'
~U,~V
denoting
the canon-
~2 ~U ~V
ical projections from
M $ U O V
that in the last case, XI,X 2
Finally, if
by
Hom(M @ U @ V,X l)
are incomparable, let
(k,I$~) , X! 0 X2) ,
~i" Then we de-
~
X
or
V, respectively. Note
Mi
or
lj.
M. or 1., we denote
1
j~
(k,(~ ~U ~V )' X @ U @ V), and by X the
(k ,
, X • U • V). With this notation, the component
•
V
of the Auslander-Relten quiver of
I2U
U
is again of one of the forms
the R-module
R-module
onto
both have to be of the form
R
containing
M
is as follows:
IIV
M V
-..I2VoI iI2-~I]U ~I]-~ V -~Ii
/
/N./
I2I 3
V --> M.->MU-~M.M^-,M~V"'"
12
13
14
/M2U
_
/
I3
/
Il
UV
M1
MI
I
/
14
12
I
M]
M2
M2
I4
13
M112
M2II
M3
~4
I
l
M2
z
:
M2M3
M
..~
/
16
15
14
• .°
to any
M212
. .
Note that in
Ii
M] 13
.
.
M311
.
M4
M5
~
,°.
MR, there is a chain of irreducible maps from any
Mj, for example a chain of length 6 from
I1
to
M 1 = M;
whereas in
MF, the modules I. and M. belong to different coml
]
ponents. Thus, two Auslander-Reiten components of M F are joined to
form, together with additional modules, a single component. Also note
that this component has the following property: there are noninvertible maps between modules of the component which cannot be expressed as sums of compositions of irreducible maps, namely all maps
from a module of the form
Mi
to a module
lj.
270
For the remaining two examples, £
,
will be the
~12-quiver
and we will consider the two simple regular modules
of period 2, with
dim U = ( 1 0 1),
dim V = (O ] O). Let
U.
U,V
be the
i
indecomposable regular module with regular socle
length
i, and similarly
V.
U
and regular
indecomposable regular with regular socle
i
V
and regular length
i. We denote by
I.
the indeeomposable X-module
i
with dimension type
(i,i-l,i-1), and by
J.
that of dimension type
i
(i,i,i-l).
First, let
by
M = V2, and consider the regular enlargement of F
M. We obtain the one-relation algebra
R
given by
C~
,
The vectorspace category
uI
Hom(MF,M F)
V2
V3
.
.
.
=
O.
is of the form
U3
U2
.
V1
~¥B~
.
14
13
12
J3
J2
Jl
V4
where we have added to a point of the form
The Auslander-Reiten component of
Il
.
R
Hom(M,X)
containing
M
the symbol
X.
is as follows
(where we use a similar notation for indecomposable R-modules as in
the previous example, and where
dimension type
13
S
denote the simple R-module of
(l 0 0 0)):
72
s
"'" ~--'*I
3 3 J2 + I - "2* ' I -2J1 ~ J 1~_ I 1
/ \/
I4~
•
/
h_7
I3Jl
"~ _ 7 t 2 N
1
13N / a 1~ /~'a /v'a /_a
U1
/ I4J~_/
/v3\ 7 h
.-
~ V ->U.l V~-*Uw'*U.V,-*~.
~_7 14 ~ / J 2 ~ /U1Jl~ / U2 ~_~v2~
2 ~ z z 4 4 "'"
I5J1
""
/ I5~ / J3~ !UIJ2~ 7U2JI~ 7 U3 ~_ / V 3 ~ /UIV4~ / 2v5
"f6
Next, let
J4
UIJ3
U2J2
U3J I
U4
M' = U I @ VI, thus the enlargement of
V4
F
by
UIV 5
"'"
M'
is the
27l
algebra
R'
given by
.or-----+
The vectorspace category
U]
U2
U3
U4
VI
V2
V3
V4
= O,
6~
Hom(M~,M F)
o l ,
ye = O.
is
14
13
12
II
J4
J3
J2
Jl
* . ,
where again we have denoted the object
X. The Auslander-Reiten component of
Hom(M,X)
R'
just by the symbol
containing
U
and
V
is
as follows:
~,
"'" 13
Vii2 "'" V2II
! 4\ / ~ / ~
llJ4
""
473
~12
IiJ~
I ~J^
/
12J I
\
/
kX
I3
~' /
"J'
\ /
J4
UIJ3
,..
3.8.
~
Y]J2
\/
/
/
~/
.J3
\ /
~\
~ 2k
U~V.
~
U2J]
U2
/ \
/
/
\ f ~ /
U2J2
/
UIV 2
U3JI
/%
U3V 3
\ /
U2V 3
.,.
V^V 4
1~ ~,~
"~/
s
UIV4 "'"
v3
U.
...
U V_
7 X /
\/
U4V 2
U2V 2
'~/
/
"'"
U.V~
!
, ~_ l-\ //
/
v2
1
\
U4VI
U3V I
UIV I
~
\ /
_/ \ /
/
S
I]
31
.,2I
I X~_/' \
~(
X
]\
I J
15
~^/
.,~ " ~ _ /
I4Jl
"'" /
7,
llJ I
\
13J 2
...
I1
I. J~
12J 2
• ,,
Vlll
Jr
k_ /
~ \ 73.
/
/
"*v
!
v4
/
/
\
...
-.-
...
Fur.ther examples of non-domestic tame algebras
With the help of Theorem 3, we can construct a large amount of
non-domestic tame algebras, and we want to mention at least some of the
algebras which arise in this way.
272
First, consider
say by
MF, thus
vertices
and
an arbitrary enlargement
MR
m
reduces
arrows,
to
then
tices, m+g arrows, and
h
geneous generators and
h
R
of
of a quiver
£,
is a quiver with
n
where
g
is the number of homorelations of
M.
with minimal projective resolution
g
Q
j =1
Pi,Qj
R
F
the number of homogeneous
M
h
@
)
J
$
i=]
indecomposable
homogeneous
If
is defined by a quiver with n+] ver-
relations,
Recall that for a module
and
M(kMF).
generators,
P.
projective,
and
h
~ + M---+
0
l
one calls
g
the number of
the number of homogeneous
relations
M.
As a consequence,
be one-relation
algebras.
ular enlargements
tion algebras.
we see that such enlargements
very seldom will
For example, we know that there are no reg-
of a tame quiver of type
(E6,2)
which are one-rela-
In writing down quivers with more relations,
the following convention:
For any pair consisting
we will use
of a black circle and
a black square with an oriented path from the circle to the square, we
have to take the relation given by the sum of all paths from the circle
to the square. Any additional relation will be given separately
examples,
(in our
the starting point usually will be one of the black dots,
the
end point a white square). Again, we do not write down the orientation
of the arrows which do not appear in relations,
obvious reflection
functors
since we can use the
in order to carry one possible orientation
into any other. As a consequence,
any of the diagrams below stands for
a certain number of isomorphism classes of algebras
field,
(for a fixed base
this number depends on the number of edges without orientation,
and the corresponding
enlargements
symmetry group). Let us write down all regular
of tame quivers with pattern of similarity
The case
(~6,2).
of a quiver of type
~6
This is the list of all regular enlargements
by a simple regular module of period 2.
Quivers with one relation:
: - o
type,S.
273
Quivers
with
two relations:
~o
B~ = 0
BI~ 1 = B2~ 2 = B3~ 3
~3
~4
o
~4~3e2 ~1 = 0
c~2c~ 1 = ~33~32B 1
Quivers
with
three
~z
relations:
0
;->_-
274
o
~-"
c~c~2 62""~
o
o
BlC~ l = 0 = 62o, 2
~6=0
One quiver with four relations:
type
The case ('A33' 1)
(E'6,2), namely
I
Besides the duals of the first algebras of
t h e r e are j u s t t h r e e a d d i t i o n a l
cases
We have seen above that there are ]9 different possibilities for
275
regular enlargements of tame quivers of type
(~6,2). Note that this
counts only the essentially different possibilities, not taking into
account the orientation of the arms. For some other pattern, let us
give the corresponding numbers of essentially different possibilities
of regular enlargements of tame quivers:
20
9
61
Next, assume that
quiver
F, and
Hom(Ms,M S)
MS
S
28
202
is the concealment of a tame connected
a regular
S-module. Then the pattern of
is the same as that of
Hom(NF,MF), where
Nr
a corres-
ponding F-module, Namely, we know that there is an equivalence
tween a cofinite subcategory
of
U
of
MF
q
be-
and a cofinite subeategory
M R . We can assume that all regular F-modules lie in
U, and then
gives an equivalence between the regular F-modules and the regular
R-modules. Thus, let
N = - 1 (M). We use now a remark in 2.3
to see that the vectorspace categories
Hom(Ms,M S)
and
in order
Hom(NF,M F)
belong to the same pattern. Thus, a classification of the tame regular enlargements of tame connected quiver (Theorem 3), immediately
also applies to regular enlargements of tame concealed quivers.
We consider just two examples of concealed quivers of type
and there the regular modules of type
o
~
o
and
~6'
(~6,2), namely
o
o
o
o
For one orientation, we write down the dimension types of the two simple regular modules of period 2.
o
o
1
:
t
t
1
1 1
and
0
1
~
0
:
0
O0
1 1 1 0
1
t 0
0
11
and
I
1 2
1 1 .
,
276
The possible extensions
by a simple regular module
of
of period 2 are as follows:
o
1
2
c~3"~°~4
°I
c~4a3a2a 1 = 0
O
Bla I = B2a 2
•
~
'
a 3 a 2 a I = O,
BlO~ 1
=
6aI
61c~1 =
= 0
B2c~2
B2
82a 2
.
The first of the cases is the dual one of an algebra considered
(namely an
(~6,2)-extension
Similarly,
for
o
o V
of a quiver),
o
~
we obtain the following ex-
tensions by a simple regular module of period 2. Always, we have
~
= O.
above
the others are new.
277
Quivers with two relations:
"%:0/'Quivers with three relations:
o
+~o-----+u
o
BT2Yl
= 0
o
-
o
Quivers with four relations:
Finally, we show that we can combine regular enlargements and
regular "co"-enlargements without much difficulty. Consider the following one-relation algebra
278
I~ :
~ O >o ) ~ _ . ~
~o----+o
F
We claim the following: Given an indecomposable R-module
Xb # O, then either its restriction
else
X
XI£
to
r
belongs to a Auslander-Reiten component
of a component of the form
(~q 13)'
X
with
is preprojective, or
C
which is the dual
discussed in 3.7. This follows
immediately from 3.7, where we have described the module category of
the opposite algebra
choose for
MR
R °p
in great detail: As a consequence,
one of the following indecomposable modules
if we
I,
U*, V *
with
0
0
o
I
diml = I 1 I
I 1 I, dimU*= 0 0 I
11
0 0 O, dimV*= 0 1 1
1o
then the vectorspace category
°.o
o
Namely, if
Hom(MR,XR) # 0
either
belongs to
XR
1
1
Hom(MR,M R)
o
i
belongs to the pattern
~
for some indecomposable R-module
C, or else
0 0 O,
O0
X
XR, then
is in fact a F-module and
XF
is preinjective. The right part of the pattern comes from the preinjective F-modules, and it is easy to see that we obtain just a chain of
objects
Hom(MR,XR)
with
XR
indecomposable in
non-domestic pattern, we consider
M × N, where
faithful representation of a A2-quiver
ments o~
M = I
R
by
~
~o
M × N
C. In order to use a
NA
is the minimal
A. The corresponding'enlarge-
lead to the following quivers with relations:
~1
~2 ~3 ~
~
~o---+e---+~--+~
~
)
>o
~5~4~3~2~i = 0
279
B~=O
M=U*
0
0
....>~0
.........>'0
YiY2 e = 0
>
Y
o
M = V*
7o
It follows
that these quivers with relations
~o
~3e2~i
= O
are tame, and non-domes-
tic. However note that in any case there is just a countable number of
indecomposable
3.9
'l'Wo
representations
In our investigation
They are not concealed
Xa # 0
of the tame one-relation
quivers,
we will
which are enlargements
algebras,
and they have to be considered
a.
two al-
separately.
since they will turn out to be nontame quivers).
that they are not enlargements
However,
for all vertices
algebras
(and there are no non-domestic
seen easily
module.
with
special one-relation
gebras were not yet touched,
domestic
X
Also,
it can be
of a tame quiver by some
see that they are specializations
of algebras
of a tame quiver by some regular module,
so that
we can use Theorem 3.
The first one-relation
algebra
R
which we have to consider
given by
O•C]"
0
0
c2
ct
:1% .....
bs- 2
is
280
We can assume that
B
is directed as
o
+o , so that a is a
a
bI
source. Namely, otherwise there has to exist a source b
and a path
m
)o ...
o
>o )o , since we have excluded oriented cycles, and
~m
bm-]
h 2 b] a
then we can apply a product
OblOb2
obtain a new orientation with
is a specialization
a
... Obm of reflections
in order to
being a source. Now we see that
R
of the following quiver with two relations
with
O
.°.
c2
~'~
O
ct
bs
bs- 1
and this is a regular enlargement
thus of type (Dn,(n-2)
@ (n-2)).
R
R
F'
of type
that
R
=
O,
S'~ = 0
P(bs)/P(bs_1)
It follows
in this way, whether
In order to see that
bI
of a quiver
n = s+t+4, namely by the regular module
cannot decide,
b2
~n' with
$ P(bl)/P(bo) ,
is tame, but we
is domestic or not.
we write
R
as an en-
largement of a quiver of finite type by some module. Let
is non-domestic,
F
be the
quiver
o ...
c2
and
M = P(bs)/P(bs_])
culate the vectorspace
a subcategory
2
o
bs
@ p(bl). Then
category
of a vectorspace
from the fact that
precisely
~
ct
)o ...
bs_ !
o
b]
M R = M(kMF),
Hom(MF,Mp).
and we have to cal-
It turns out that this is
category with pattern ~ ,
is a subquiver of
the regular r'-module considered
ry is itself non-domestic.
o
b2
F'
and
above),
(this follows
M, as F'-module,
Let us give the calculation
in one example:
Let
R :
and
M
~
~
, thus F = ~ <
is the direct sum of the indecomposable
dimension
type
quiver of
F
0 1 | 0 0 0
is as follows
and
0 0 0 0 0 !
is
and this subcatego-
o---+o(--o----+o ,
representations
of
The Auslander-Reiten
281
rrlFj °t H
and we have indicated both
Hom(M,M r)
Hom(~11OOO,MF)
and
Hom(~OOOOI,MF),
thus
is
o
We have to invoke now the classification
of the subspace of such a
vectorspaee
[29], in order to see that
category which follows from
there are infinitely many series,
The second algebra
S
thus it is non-domestic.
which we did not consider yet is
b1
b2
b3
cI
c2
c2
-
i;
bs- 1
where again we can assume that an orientation
Now we see that
S
is a specialization
is choosen with
of the quiver with two
o~ B
a
o.
bl
282
relations
a
with
!
o
bs
bs_ ,
b2
This is a regular enlargement of a
type
(~n,(n-2) @ (n-2)). Thus
we can write
S
by t h e module
O
al~l = ~;~2 '
B' ~ =
O
o
~n-quiver, where
n = s+4, of
is tame. As in the previous case,
as an enlargement of a quiver of finite type, namely,
O
c2
S
b,
~o
b
s
...
Us ,
0 '~
O
b2
b
N = (P(Cl) ~ P ( c 2 ) ) / P ( b s )
~ P ( b l ) . For example, in the
case of
S :
we obtain the same quiver
F
as in the special case of
R
considered
above, this time N is the direct sum of ] ] O O O O and
O
I
O O O O I. Let us indicate again the vectorspace categories
O
Hom(~lOOOO,M F) and Hom(~OOOOI,MF)
in the Auslander-Reiten quiver
283
As a consequence, the vectorspace category
Hom(NF,M F)
is
and c o n s e q u e n t l y , n o n - d o m e s t i c .
In o r d e r to s ~ w t h a t
and R o j t e r
S
i s tame, we have r e f e r r e d
[ 2 9 ] . The main example and t h e s t a r t i n g
to Nazarova
point for the
t h e o r y d e v e l o p p e d i n t h a t p a p e r was t h e q u i v e r w i t h r e l a t i o n
~l
~2
~
~i~i
1
,
= ~2a2 ,
2
the determination of its indecomposable representations having been
posed before as a problem by Gelfand. Note that this quiver with relation is a specialization of the situation above: we have to s ~ i n k
arrows outside the commutative square.
all
284
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Claus Michael Ringel
Fakult~t fNr Mathematik
Universit~t Bielefeld
D-4800 Bielefeld I