232
HEREDITARY ARTINI~J~ RINGS OF FINITE REPRESENTATION
P.Dowbor,
Recall
of f i n i t e
[11,6]
that
type i f
C.H.Ringel,
a hereditary
and o n l y i f
TYPE
D.Simson
finite
dimensional
a corresponding
algebra is
diagram i s
a disjoint
u n i o n of t h e Dynkin diagrams A n , Bn, Cn, Dn, E6, E7, E8, F4, G2
occouring
in Lie theory.
of a h e r e d i t a r y
It
artinian
Here, we w i l l
r i n g A and a s s o c i a t e
t u r n s out t h a t A i s of f i n i t e
disjoint
c o n s i d e r the g e n e r a l case
type if
a diagram
and o n l y i f
C(A)
F(A).
i s the
union of the C o x e t e r diagrams A n , Bn ( = Cn), Dn, E6, E7,
ES, F4, G2, H3, H4a I2~ p)
which c l a s s i f y
groups [ 3 ]
. However, the e x i s t e n c e
I2(p)
p = 5 o r p ~ 7 remains open: i t
with
of
difficult
questions concerning division
we d e f i n e
for
the root
to i t
any C o x e t e r diagram
the i r r e d u c i b l e
r i n g s of type H3, H4 and
depends on r a t h e r
rings.
artinian
ring
On the o t h e r hand,
" b r a n c h system" which g e n e r a l i z e
systems of the Oynkin d i a g r a m s .
of a h e r e d i t a r y
Coxeter
of f i n i t e
The dimension
representation
types
type j u s t
form such a branch system.
The r e s u l t s
of s e c t i o n s
1, 2 and 5 were o b t a i n e d by P.Dowbor
and O.Simson who announced p a r t
of them i n the papers ~9, 101]
and at the Ottawa C o n f e r e n c e 1979. The r e s u l t s
independently
were o b t a i n e d
by C . M . R i n g e l who announced them at the O b e r w o l f a c h
meeting on d i v i s i o n
rings
1. Bimodules o f f i n i t e
i n 1978.
representation
Let F and G be d i v i s i o n
~(FMG ) t h e c a t e g o r y of f i n i t e
a representation
rings,
type
and FMG a b i m o d u l e .
dimensional
b e i n g of the form
Denote by
representations
V = (XF,
of FHG,
YG o ~ : X F~FMG --~YG )
w i t h d i m e n s i o n t y p e dim V = (dim XF, dim YG). Given FMG, l e t
HR = HomG(FHG,GGG),
ML(i+l):
HL : HomF(FNG,FFF) , and ~ # ( i + I ) =
(MLi) L, w i t h
HR° = M = ML ° .
Note t h a t
if
(MRi)R,
dim HG i s
finite,
233
~LE
Ht whereas i f
dimFH
is
i 8 a bimodule w i t h
finite
dualisation
ML i
are f i n i t e
Proppsitign
finite
0 then
d i m e n s i o n a l on e i t h e r
1. Assume
EMG
0
)
Pi-1
The bimodule
as a r i g h t
H
vector
9re ( s e m l l i n e a r )
bimodules
representation
P I ' " . . . . Pm" Let
and assume we have choosen an o r d e r i g
orderin 9 is unique,
all
with
type, with
dim Pi = ( x i '
ML(i-1) ~
has f i n i t e
Pi
)
dualisation;
Yi ) '
x i / Y i ~ X i + l / Y i + 1. Then
an d theE9 are A u s l a n d e r - R e i t e n
)
NR i and
side.
i s of f i n i t e
indecomposable r e p r e s e n t a t i o n s
this
if
HLR~ M. We say t h a t
Pi+l
HL i
sequences
)
O
is one-dimensional
space, f o r some i~ and the b i m o d u l e s
N
and
MLm
isomorphic.
Let o u t l i n e
the main s t e p s of the p r o o f :
If
is
dim ~
as u s u a l E 2 , 7 ] :
finite,
given
we d e f i n e
(XF~Ft4 G
~
a functor
C~:E(FH G) - - - ~ ( H L)
YG ), l e t
AG = k e r ~ .
Using
X F@FHG ~ Hom(ML, XF), we get from the k e r n e l map Ag ---)Hom(M L, XF)
as a d j o i n t
the f u l l
a map of the form
subcategory
~I(FF1G )
simple projective
direct
c a t e g o r y ~ 2 ( H L)
of
direct
A G ~ HL - - ~ X F. C l e a r l y ,
of
~ ( H L)
of o b j e c t s
~ ( H L)
XF " ~
(XF®FMG
without
and
simple injective
~H)
"~
YG)
define
dimFH
o n t o the c o k e r n e l of the a d j o i n t
or
i s of f i n i t e
N
map
an e q u i v a l a n c e
and ~ I ( H R ) .
dim HG
be of bounded r e p r e s e n t a t i o n
show t h a t
have the same
C;:~(FHG) -'-)~(HR)
H°mG(FHG ' YG ) ~ YG ~ HR' and t h u s e s t a b l i s h i n g
between . ~ 2 ( M )
If
to the fuZZ sub-
type.
SimiZary. if dim(MR~ is finite,
mapping
~1'
of o b j e c t s w i t h o u t
summands i s e q u i v a l e n t
summande. I n p a r t i c u l a r ,
representation
~(FHG )
under
is infinite,
type.
Thus, the f u n c t o r $
has t o be a bimodule w i t h
representation
type.
then c l e a r l y
finite
~(H)
~+
Ci
dualisation
cannot
and
if
~i"~tt)
234
The functors
indecomposable in
in which case
have nice properties: I f
C 1+ , C[
~(H)
, then its image in ~ ( M R)
[XF" YG' q) =(FF' O, o )
dim C~(X, Y, ~)
: (y,
yb-x),
where
b = dim(MR)F . I n p a r t i c u l a r ,
~(M)
of type ( x , y )
~ ( M R)
of type
Let
yb-x)
Call
C~(X,y,~) : 0
f o r some
i.
(X, Y, ¢ )
di..~m( X , Y , ~ )
and o n l y i f
C~:~M) ~
in
f o r some i ,
Obviously,
is simple injective, or else
: (x,
y)
and
t h e r e i s a unique one i n
~(M)
~HR~
be the i t e r a t e d
preprojectiv e
provided
provided C~(X,Y,~) :
and p r e i n j e c t i v e
these modules are c h a r a c t e r i z e d
Pi
with
Ci-l+ Pi # O. Being d e t e r m i n e d by t h e i r
and
dimension types,
the l a s t
Pj
be the indecomposable module ( i f
can be i s m o r p h i c o n l y f o r
such module which e x i s t s
representation
Pm
Pi'
type).
Then a l l
have to be i n j e c t i v e .
(assuming
Pi
i=j.
with 1 ~ i
5 m exist,
c l o s e d under indecomposable submodules of d i r e c t
modules, i t
it
exists)
Let
Pm
be
.~(M ) of f i n i t e
Since the s e t of p r e p r o j e c t i v e
the indecomposable i n j e c t i v e
o
by t h e i r
d i m e n s i o n t y p e s . Let
C~z Pi = 0
in
# (0,1).
C+:Z(["1)i ~-~Z(r'IL~ ,
functore
if
is
is either zero.
t h e r e i s a unique r e p r e s e n t a t i o n
# (1,0)
(y,
(XF' YG' ~ )
and
Pm-l'
modules i s
sums, and c o n t a i n s
contains
all
indecomposable
modules. Similary, all indecomposable modules are also preinjective.
N e x t , one proves t h a t
Hom(P1,P2) ~ FHG , and t h e r e f o r e
ML(i-1)
Hom(Pi,Pi+&~ :
. A l s o , t h e r e i s an o b v i o u s e x a c t secuence
0
)
Pi-i
) H°m(Pi-l'Pi ) L ~
Pi
> Pi+l ~
0 ,
and it is left almost split [1] Cprove it by induction using the
functors
c[).
Similary,
and
Ci_ 1 I i
let
be the indecomposable module w i t h
~ O. Then
we c o n c l u d e t h a t
Finally,
Ii
if
Hom(i2,ii)
~ [.~2. Since
ML(m-2) ~ I ~ 2, t h u s
the r i g h t
C[ I i = 0
Pm-& = I 2 '
Pm = I I '
MLm ; H.
d i m e n s i o n of a l l
HL i
would be ~ 2, then
235
one p r o v e s by i n d u c t i o n
that
on
di..._mmP1 =(xi" Yi )
i,
that
with
Pi exists
~or
any
i
~ 1 )
and
0 ~ x i < Yi " Thls produces a
countable number of isomorphism classes of indeoomposabls
modules.
.2.. D i m e n s i o n sequences
A sequence
a i ~ tN
xi'
is
a = (a 1 ......
called
Y i 6 IN ( 1
a
-< i
=
-I,
vectors
of length
d i m e n s i o n sequence
conversly,
aiYi = Yi-1 + Yi+l
a. Note
with
there exists
( i .< i < m),
that it is uniquely determined by
it determines
a.
This generalizes
of the usual rank 2 root systems:
are j u s t
provided
= m~ 2
= O, x m = i, and Yo = O, Yl = I, Ym = O. The set of
x I xi
system
Yi / , with I --< i ~ m, is called the branch
"'
defined by
(0,0),
l al
~< m), w i t h
aixi = xi-I + xi+l '
x0
am)
(1,1,1),
(1,2,1,2)
the p o s i t i v e
and
roots
the positive part
for the dimension sequences
(1,3,1,3,1,3)
of
a, and
the b r a n c h s y s t e m s
A 1 x A1, A2, R2
and
G2~ r e s p e c t i -
vely.
Proposi_tion, 2 .
The branch s y s t e m i n
follo ,s: {(I,o>, (o,i) I
system,
and.
p,q
iR2
can be c o n s t r u c t e d
is a branch system,
a r e neig, h b o r s i n ~
as
is a br nch
, then ~ v ~ p + q t
is
a branch
system.
Here, in a finite subset
linearly intependent
-~
of
IR2
consisting of pairwise
elements, we call two elements neighbors in
case the lines through these elements are neighbors in the set of
all lines through elements of ~ .
G. Bergman has pointed out that
the branch systems
~(x i, Y±~ I I 5 i < m I Correspond just to the
x.
Farey sequences
~
(see [16]). In particular, the numbers
Yi
xi' Yi always are without common divisor.
The proposition above can be reformulated as follows.
236
Proposition
2~
as f o l l o w s :
The s e t
(0,0)~
~
of d i m e n s i o n sequences can be o b t a l n e d
, and i f
sequences
(al,...,ai_l,
1 ~ i < m
belon~
( a l , . . . . am) ~
at+l
, 1, a i + 1 + 1 ,
some
cyclic
sequence
i ~ 2. An easy consequence i s
Corollary,
ai+ 2 .....
the
am )
fo_._r
to,~.
Note that for any other dimension
for
, then a l l
The set
~
a, we have
ai = 1
the following
of d i m e n s i o n sequences i s c l o s e d under
permutation@.
Given a d i m e n s i o n sequence
= (am,a I . . . . .
am.l).
a = (al,...,am),
Let us g i v e the l i s t
of l e n g t h ~ 7, up to the c y c l i c
CO,0);
(1,1,1);
of a l l
permutations
(1,2,1,2)$
~1,2,2,2,1,4),
a
+
=
dimension sequences
and r e v e r s i o n :
(1,2,2,1,3);
(1,2,3,1,2,3),
(1,2,2,2,2,1,5),
let
~1,3,1,3,1,3);
(1,2,2,3,1,2,4),
(1,2,3,2,1,3,3),
~1,4,1,2,3,1,3).
3 t C o x e t e r diaqrams
Let
d: I~
I~
oriented
With
P : ~1 . . . . ,n }
---)~.
Note t h a t
graph,
if
(P,d)
IP , J d l ) ,
and assume t h e r e i s g i v e n a s e t map
d
d e f i n e s on
we draw an arrow
we a l s o will c o n s i d e r
where two d i f f e r e n t
i -->j
i n case
points
d e f i n e d by) d, d e f l n e
i,
as follows: i f
( ~ i x ) j = x3
If
for
il,...,i
# (0,0).
$ 3.
If
(~id)Ci,J)
i
is
a sink for
= d ( 3 , i ) +,
for a l l
a l i n e a r transformation
j,
k
# i.
Ei = ECd)i
x = Cx j ) ~ IRn, l e t
for
t
define on IRn
d(i,j)
are c o n n e c t e d by at most two
(]Tid) ( j , i I = { 0 , 0 ) , and C~id ) Cj,kl = d ( j , k )
For any sink
of an
t h e u n o r l e n t e d graph g i v e n by
edges, and any edge has a s s i g n e d a number
~the o r i e n t a t i o n
C the s t r u c t u r e
j # i,
is a
and
( ~ i x ) i = -x i +~'dCi'J)li x j .
(+)-admissible sequence "(thus
d, for a l l
I
s
t),
define
is
i s a sink
237
~-it...i 1 = ~(~t.l...~d)it
We c a l l
y e PNn
l e sequence
canonical
il,...,i
t
Theore.m 1.
disjoint
of
i s of f i n i t e
O2 , H3, H4, I2~P) ~p=5
C o x e t e r diagrams of
or
rank
system;
12(P) , H3~and
•
~)
contains
and i $
An, Bn ( = C n ) ,
If
and w i t h
i s of
the
H4
Bn,or
have
let
• , with
the f o l l o w i n g
½ nh
i s one of the
h,
then t h e
elements.
A n, On, E6, E7,
r o o t s of the c o r r e s p o n d i n g
branch systems f o r
Bn, namely
Cn. The branch systems of the t y p e
p, 15, and
us w r i t e
60
elements,
respectively.
down the branch system f o r
second d i m e n s i o n sequence
15
(2,1,3,1,2).
It
vectors
~1 0 0~,
(0 1 0 ) ,
(0 0 1 ) ,
C1 1 0 ) ,
(0 1 1 ) ,
[1 1 1~,
(0 I 2 ) ,
CO 2 1 ) ,
C1 I 2 ) ,
(1 2 1 ) ,
(1 2 2 ) , ~2 2 1),
~1 3 2 ) ,
(1 3 3),
(1 4 2),
dependent
i $ the
Dn, E6, E7, E8, F4,
t h e diagrams
the p o s i t i v e
t h e r e are two p o s s i b l e
r o o t s of
(C , I d [ /
(C , ~ d l )
C o x e t e r number
the branch systems f o r
As an example,
~
vectors
and o n l y i f
has precisely
are p r e c i s e l y
the p o s i t i v e
-
type i f
p ~ 7).
n
branch system for (P,d)
root
(r ,d)
many p r e p r o j e c t i v e
the s e t o f p r e p r o j e c t i v e
union of C o x e t e r diagrams
E8, F4, G2
Vie say t h a t
~1~ , d ) .
(r,d)
Of c o u r s e ,
0).
a (+)-admissib-
i s one of the
t h e r e are o n l y f i n i t e l y
and then we c a l l
•
there exists
~i t...il(x)
CO, . . . . 1 , 0 . . . . .
provided
branch s~stem
provided
such t h a t
base v e c t o r s
f i n i t 9 .t.ype
vectors,
~feProjective
o\t_l...il
on t h e o r i e n t a t i o n
and the g i v e n dimension
sequence.
The p r o o f of the theorem i s
constructs . x p l i c i t y
rather
for every pair
technical.
(r,d)
First,
with ( r , l d l )
one
e Coxeter
diagram the corresponding branch system. For the converse,one only
238
has t o c o n s i d e r
of
(C,
Idl)
the
pairs
(F,d)
are C o x e t e r
shows t h a t
in
for
diagrams,
thmse c a s e s
there
wich
all p r o p e r
and a g a i n
subdiagrams
an e x p l i c i t
are i n f i n i t e l y
calculation
many p r e p r o j e c t i v e
vectors.
4. Here,dltar,? artinian rin.qs
Given an artinian ring
A, let
A°
n
=
T'T F i
, and
radAO/(radA°)2
=
~
i=l
be its basic ring,
n
iMj
as
~-r
i,j
= (Fi,iMj)
is
called
dimension
sequence o f
A
(r,d)
a pair
Theorem 2.
the species
the bimodule
grams
Then
of
A.
Let
iHj
, t h u s we have a s s o c i a t e d
d(i,j)
the pair
The hereditary artinian rin~
ratio ~ type if and only if
Fi-bimodu]~.
i=I
r(A)
and we d e n o t e b y
A°/radA °=
A
be t h e
to
(P,Idl).
±_s of finite represeqr
i~(A) is disjoint union of Coxeter dia~
An, B n (=Cnl , Dn, E6, ET, E8, F4, G2, H3, H4, I2(P) [p=5
9L
p ~ 7). In this case r the dimension types of the indecomposable
A-modules form the branch system for
[1~,d).
As in the case of an algebra L6], one sees that for a hereditary
artinian ring
A
of finite representation type, the basic ring
A°
always is the tensor ring over its species.
5.
P r o b l e m s on d i v i s i o n
rinRs
The main p r o b l e m w h i c h a r i s e s
answer is
the question
bimodules
a2 = &
some
end
FHG
of
finite
~uaing a cyclic
MLi).
Then t h e r e
FMG = FFG . S i n c e
starts
about
the
and w h i c h we are n o t
possible
dimension
representation
permutation
is
given
of
type.
a,
a division
able
to
sequences
a
We may assume t h a t
thus
replacing
M
ring
inclusion
G c--~F,
HL = GFF , t h e d i m e n s i o n
sequence of
by
FFG
as follows:
a 1 = dim F G , a 2 = 1 , a 3 = dim GF .
Thus,
left
immediatly,
or
right
we are c o n f r o n t e d
index
of
a division
with
subring
the question
G
of
of differrent
F ,(see
[4]~
of
239
Of p a r t i c u l a r
blmodule with
fact,
since
it
is
the question
the d i m e n s i o n sequence of
(2,1,5,1,2),
In
interest
is
it
would give
rise
e a s y t o see t h a t
(2,1,5,1 ....
with
dim F G = 2 , d i m GF = 5 , and
L e t us p o i n t
length
of
)
is
~2,1,5,1,2)
again isomorphic
to
5,
Also
F
Bimodules with
esttng
there
examples of
exists
course,
(this
FHG
of
to
[13]
these conditions.
follows
from
R
Similary,
follows
that
F , G
subring
H
F o r e x a m p l e , we may ask w h e t h e r
finite
representation
from proposition
one-dimensional
3 , and w i t h
with
rings
F , G
ring
ring
with
type with
,am).
1,
since
Of
as a
dualisation
on one s i d e ) .
Note t h a t
have t o be i s o m o r p h i c ,
extension
R :
F K M ,
(radR) 2 = 0 , left
precisely
(radR) 2 : 0 , left
m-1
length
indecomposable modules.
type
length
I2(5 )
would give
3 , right
length
a
3 ,
4 indecomposable modules.
a bimodule with
= u , a2 :
ai
Since the
ML5 ~ H
a division
the dimension sequences of
and p r e c i s e l y
remaining
dlmFHOm(GFF,GG) = a 4 = 1 .
a l s o can f o r m t h e t r i v i a l
ring
G ~ F
such a b i m o d u l e has t o be s i m p l e
t h e n w o u l d be a l o c a l
aI
T h u s , we w o u l d need
rings.
and t h i s
where
d i m e n s i o n sequence s t a r t i n g
contains
odd m, t h e d i v i s i o n
In particular,
H4 .
artinian
also
length
and
dimension sequences would produce inter~
we
local
H5
sequence ( 3 , 1 , 2 , 2 , 1 ) .
so t h a t
3 , right
type
dim F H = S , dimHF = 2 , s i n c e
leads to a bimodule which is
i n case o f
a
say
dim MG , t h u s d i m e n s i o n s e q u e n c e ( 2 , 2 , a 3 . . . .
according
bimodule
of
exists
suitable
a bimodule
dimFM = 2 :
F
with
(FFG)L3 has the dimension
it
I2(5),
rings
consequences of
is
have t o be i s o m o r p h i c .
to
(2,1,3,I,2)o
out certain
type
the only
with
whether there
= 2
d i m e n s i o n sequence
1 , au+ 1 = 3 , au+ v = I
w o u l d be o f
local-colocal
(a 1 .....
, au+v+ 1 = v
representation
au+v+ 1 )
, and t h e
type
( a n y i n d e c o m p o s a b l e m o d u l e s has a u n i q u e m i n i m a l s u b m o d u l e o r u n i q u e
maximal submodule).
with
this
ProPerty
In contrast,
all
the
finite
have been c l a s s i f i e d
dimensional
in
[15].
algebras
240
T h i s shows v e r y c l e a r l y
t h e o r y of a r t i n i a n
the dependence of the r e p r e s e n t a t i o n
r i n g s on q u e s t i o n s c o n c e r n i n g
f a c t which to have been e x h i b i t e d
In p a r t i c u l a r ,
division
finite
as
rings
for
the f i r s t
under the assumption t h a t
F D G
with
any a r t i n i a n
time i n
a
[14].
of
dim FG = 2 , but
H3, H4, and I 2 ( P )
ring
rings,
t h e r e are no p a i r s
dimGF = 2 , and
, the C o x e t e r diagrams
P(A)
for
division
cannot be r e a l i z e d
A , so t h e y c o u l d be e x c l u d e d from
theorem 2.
Of course,
in case we assume
that the division
finitely generated over their centers,
the cases
H3,H4, and I2(p),
rings
then we exclude
F
are
immediately
thus one has the following (see also
[8]):
The
Theorem 3.
finitely
if
hereditary
generated over its
and o n l y i f
A n , Bn ( = C n ) ,
r(A)
is
artinian
center is
the d i s j o i n t
rin 9
of f i n i t e
A
with
A/radA
representation
typ~
union of O o x e t e r diagrams
Dn, E6, E7, E8, F4, G2.
Re,ferences
[1]
Auslander,
H.,
algebras
(1975),
[2]
Berstein,
Reiten,
III:
I.:
R e p r e s e n t a t i o n t h e o r y of a r t i n
Almost s p l i t
239-294.
I.
functors
N.,
Gelfand,
I.
and G a b r i e l ' s
(1973),
Bourbaki,
Paris:
[4]
H. Ponomarev, V. A . :
Russian Hath. S u r v e y s 28
17-32.
No: Groupes e t
Hermann
Cohn, P. H . :
a l g e b r e s de L i e ,
Dlab, V.,
Skew f i e l d
Ringel,
uniserial
Ch. 4 , 5 , 6 .
1968 .
constructions.
L e c t u r e Note S e r i e s 27~ Cambridge
[5]
Coxeter
theorem. U s p i e c h i H a t . Nauk 28,
19-33 ( 1 9 7 3 ) , t r a n s l a t i o n :
[3]
sequences. Comm. A l g e b r a 3
C. N . :
rings.
London Hath. Soc.
1977 .
D e c o m p o s i t i o n of modules o v e r r i g h t
Math. Z. 129 ( 1 9 7 2 ) ,
207.230.
241
[6]
-,-
: On a l g e b r a s of f i n i t e
(1975),
67]
-,-
representation
t y p e . U. A l g e b r a 33
306-394.
: IndecoBposable r e p r e s e n t a t i o n s
of graphs and a l g e b r a s .
Memoirs Amer. Math. Soc. 173 ( 1 9 7 6 ) .
[8]
Dowbor, Po, Simeon, D , : Q u a s i - a r t i n
finite
[9]
-,-
representation
i1~
-,-
[1~
Gabriel,
-
type.
of h e r e d i t a r y
~973),
C. M.: R e p r e s e n t a t i o n s
41
~976),
Rosenberg, A . , Z e l i n e k y ,
injective
[1~
hull.
preprint.
Nanuscripta Hath. 6
II.
$ympoeia Hath. I s t .
81-104.
of K - s p e c i e s and b i m o d u l e e .
296-302.
D.: On the f i n i t e n e s s
H a t h . Z. 70 ~ 9 5 9 ) ,
of the
372-380.
Tachikawe, H . : On a l g e b r a s of w h i c h e v e r y lndecompoeable
representation
has an i r r e d u c i b l e
bottom Loewy c o n s t i t u e n t .
[1~
I.
type.
71-103.
9. A l g e b r ,
[14
representation
: IndecoBpoeable r e p r e s e n t a t i o n s
Ringel,
r i n g s of f i n i t e
P.: Unzerlegbare Darstellungen
Naz. A l t a H a t . V o l . XI
[13]
to a p p e a r .
to a p p e a r .
: On b i m o d u l e s of f i n i t e
(1972),
[14
t y p e . J. Algebrae
: A characterization
representation
s p e c i e s and r i n g s of
Vinogradov,
I.
one as the t o p or the
Hath. Z. 75 ~ 9 6 1 ) ,
M.: Elements of number t h e o r y .
215-227.
Dover
publications ~ 9 5 4 ) .
P. Dowbor , D. Simeon
C. H. R i n g e l
Institute
Fakultat fur 14athematik,
of M a t h e m a t i c s ,
Nicholas Copernicus University,
Univereitat,
87-100
D-4800
Toru~
Bielefeld