A SPECTRAL SEQUENCE APPROACH TO NORMAL
FORMS
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
1.
Introduction
Normal forms for vector elds and Hamiltonians at equilibria have
a long history, an extensive literature, and a continuing appeal for
researchers (e.g., see the references in [Mur1], [Sa ]). These entities
have been treated in terms of completions of graded Lie algebras for at
least 40 years [C], and more recently, following [B], in terms of actions
of a graded subgroups acting on that Lie algebra.
The group action context allows for a very simple description of the
normal form problem: nd the orbit representatives which in some
sense are the smallest. Baider characterized such elements in terms of
a decomposition of the Lie algebra involving the image of the action
and a complement; the minimal representative of an element is the
one which lies entirely in the part that cannot be killed by the group
action, and that representative is unique [B].
It has been known for quite some time that the standard methods
for computing normal forms in the graded Lie algebra setting are related to spectral sequence calculations (see Arnol'd [A] for the case of
singularities; Sanders and Murdock [Sa ], [Mur1] for the case of vector
elds). Specically, in [Sa ] Sanders showed how one could interpret
the normal form algorithm in terms of a minor variation of the standard
spectral sequence of a ltered module with a compatible grading (also
see [Sa ]). These spectral sequences provide some valuable information
about the normal form but do not seem to play a major role in the
actual calculations. Here we generalize the normal form algorithm to
situations not covered by [B] and use a dierent approach to construct
spectral sequences indexed by the elements `. This approach allows
1
1
1
2
Date : December 13, 2004.
Key words and phrases. Spectral Sequence, Normal Form.
1
2
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
us to compute the normal form entirely in the context of the spectral sequence and to construct morphisms between spectral sequences
indexed by elements in the orbit of a group action.
Our constructions can be viewed in terms of a category OC associated with each orbit O of a group action ' : G X ! X : the objects
are the points of the orbit; a morphism between objects X ; X is
an element g 2 G such that g X = X ; composition is dened by
multiplication. When the action is initially linear, as dened in x5,
and when one additional hypothesis is satised, we construct a functor
from OC to a category of short cochain-complexes, thence to the category of spectral sequences. We then show that the resulting spectral
sequences are invariants of the given orbit, i.e., that all are isomorphic
(see Theorem 6.11), and that the calculations involved in computing
this spectral sequence include those involved in calculating the normal
form.
Section 2 establishes notation, and x3 and x4 summarize standard
material. Specically, x3 is included for the benet of normal form
workers with no background in spectral sequences, and x4 is for those
spectral sequence workers unfamiliar with normal forms.
Sections 5 introduces the notion of an initially linear map and generalizes normal form theory to the action of a group on a vector space.
This goes beyond Baider's context and encompasses other widely studied \normal form" problems, e.g., matrix normal forms as in [GR].
Indeed, to keep the calculations from becoming unwieldy we stick to
matrix examples. In x6 the actual spectral sequences are introduced.
Our methods also apply to cyclically graded Lie groups. In particular, we are now able to treat the one normal form case for indecomposable linear Hamiltonian operators which could not be handled using
the methods developed in [CK]. This work will appear elsewhere.
The paper should be regarded as an application of homotopy theory,
in the guise of elementary spectral sequences, to problems in analysis.
Although far aeld from the lecture delivered by the rst author at the
conference celebrating Sam Gitler's 70th birthday, it seems a tting
illustration of the rich diversity of Sam's interests.
1
1
2
2
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
2.
3
Preliminaries
Throughout the paper R denotes a commutative ring with multiplicative identity 1 6= 0 and all modules are assumed (left) R-modules
unless otherwise stated. A ltration fF pM gp2Z of a module M (by
R-submodules F pM ) will always refer to a decreasing ltration, i.e.,
(2.1)
)
q>p
F qM
F pM:
When the inclusion in (2.1) holds we refer to F q M as a higher ltration
than F p M .
We will always deal with modules M having the following structure: fMp gp2Z is a family of free modules of nite dimension, F pM :=
Q
p
q p Mq for each p 2 Z, and M := [p2ZF M . The construction
guarantees that elements m 2 M can be regarded as formal innite
sums
(2.2)
m = mq + mq + +1
with
m p 2 Mp ;
which for q < 0 one could think of as a Laurent series. Note that
fF pM gp2Z denes a ltration of M . We refer to such modules as
(Z -)graded modules. (This is a mild abuse of standard terminology:
graded objects are generally assumed direct sums, whereas M lies
L
Q
between the direct sum p Mp and the (direct) product p Mp .)
For any p 2 Z the p-jet Jp(m) of m = mq + mq + 2 M is
dened by
mq + + mp if p q
(2.3)
Jp (m) :=
0 otherwise:
When a graded module M is also Lie algebra with bracket [ ; ]
satisfying
+1
(2.4)
[Mp ; Mq ] Mp
+q
for all
p; q 2 Z
we refer to M as a (Z -)graded Lie algebra. When this is the case and
m 2 M we let ad(m) : M 7! M denote the standard adjoint mapping
n 2 M 7! [m; n] 2 M . We use brackets to denote cosets of submodules:
if a 2 M and N M is a submodule we write a + N M as [a]
and say that a represents [a].
4
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Examples 2.5.
(a) Fix an integer n 1 and let L := TU (n; R) denote the Lie
subalgebra of gl(n; R) consisting of the upper triangular matrices.
(The bracket is the usual matrix commutator [A; B ] := AB BA.)
L
One can view L as having both the direct sum i Li and product
Q
i Li forms by taking Li to be those matrices (mpq ) satisfying
mpq = 0 if q p 6= i, i.e., the only non-zero elements are on the
ith -superdiagonal, with the understanding that this refers to the
zero matrix when jij n. Condition (2.4) is easily veried. As
an illustration of jets: the 2-jet of an element
0
B 0 B
m=B
B 0 0 @ 0 0 0 0 0 0 0
is given by
0
1
C
C
C
C2L
A
1
0 0
B 0 0 C
B
C
C;
0
0
J (m) = B
B
C
@ 0 0 0 A
0 0 0 0 2
wherein corresponding entries in m and J (m) indicated by asterisks are identical.
(b) Let K = R or C and let Vect(n) denote the K -space of forP
@
in equilibrium at 0, i.e., the
mal vector elds X = nj pj
2
@xj
formal power series coeÆcients pj 2 K [[x ; : : : ; xn ]] are without
constant terms. Vect(n) is given the structure of a K -Lie algeP
@
bra by dening the bracket of elements X = nj pj
and
@xj
P
@
Y = j qj
to be
@xj
=1
1
=1
[X; Y ] :=
@q
(pi j
@xi
X
X
j
i
!
@p
@
qi j )
:
@xi @xj
It is given the structure of a graded Lie algebra by setting Vecti (n) :=
P
@p
0 when i < 0 and letting Vecti (n) denote those X = nj pj j
=1
@xj
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
5
in which the pj are homogeneous polynomials of degree i+1 when
i 0.
The study of vector elds at equilibria is one of the standard
applications of normal form theory (see, e.g., [Mur1] and [Sa ]).
1
6
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
3.
Background on Spectral Sequences
References for this introduction to spectral sequences are [Gode],
[Mac] and [Sp].
A dierential object consists of a module E together with an Rlinear mapping d : E ! E , known as the dierential, satisfying d = 0.
Any cochain complex
2
! Eq
(3.1)
1
! E q Æ! E q ! Æq
1
q
+1
L
can be considered a dierential object: take E := q E q and dene
P
P
d : E ! E by q eq 7! q Æ q eq . Indeed, alternate notation for (3.1),
which we immediately adopt, is
! Eq
(3.2)
1
d
d
!
Eq !
Eq ! :
+1
Similarly, any chain complex may be considered a dierential object.
L
p;q
Another important example is the direct sum E :=
of
p;q E
R-modules indexed by Z Z together with a dierential d : E ! E
satisfying djE : E p;q ! E p r;q r for all p; q . In this instance
the dierential object is called a bigraded module with dierential of
bidegree (r; r + 1) (e.g., see the spectral sequence charts in Example
3.18).
The derived module H (E ) of a dierential object (E; d) is dened
by
(
p;q
+
)
+1
H (E ) := kerfd : E ! E g=dE ;
(3.3)
this module is also called the cohomology (resp. homology) of E , particularly in the case of a cochain (resp. chain) complex.
A spectral sequence is a sequence f(Er ; dr )g1
of dierential objects
r
such that Er ' H (Er ) for all r. In the latter denition no relationship between the various dierentials is assumed, although in practice they are often induced by the same mapping. We follow custom
and express the R-module isomorphisms Er ' H (Er ) as equalities.
Moreover, when confusion cannot otherwise result we write all dr and
all restrictions thereof as d.
=0
+1
+1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
7
A map (or morphism) f : f(Er ; dr )g1
! f(E r ; dr )g1r of spectral
r
sequences is a collection of R-linear mappings fr : Er ! E r commuting with the dierentials, i.e., satisfying fr Æ dr = dr Æ fr for all
r 0.
Suppose f(Er ; dr )gr is a spectral sequence and k 0 is an integer.
An element e 2 Ek survives to Ek if e 2 ker dk , in which case e
determines a coset [e]k 2 Ek = H (Ek ). Inductively, e survives
to Ek n if it survives to each Ek r with 1 r < n and each [e]k r
is in the kernel of dk r . The notation [e]k r is somewhat misleading
given our bracket convention for cosets: the coset [e]k r of [e]k r in
Ek r is seldom represented by e (as we will see in examples). All we
can say is that [e]k r is represented by an element with leading term
e in lowest ltration. An element e 2 Ek is killed if e 2 dEk . Notice
from dk = 0 that such a class must survive to Ek and represents 0.
We will only be interested in spectral sequences f(Erp;q ; dr )gr of
bigraded modules with dierentials dr of bidegree (r; r + 1). Such
a spectral sequence strongly converges if for each (p; q ) 2 Z Z there
is a non-negative integer r(p; q ) such that dr jE is the zero homop;q
morphism whenever r r(p; q ); the denition E1
:= Erp;q is then
independent of r r(p; q ) (up to isomorphism) (see [Sp, page 467]).
A spectral sequence as in the previous paragraph is a j th -quadrant
spectral sequence if Erp;q is the trivial module whenever the pair (p; q )
is not in (the closed) quadrant j; j = 1; 2; 3; 4.
A collection of subcomplexes
=0
=0
0
+1
+1
+1
+
+
+
+
+
+ +1
+
+ +1
+ +1
2
+1
0
p;q
r
! F pE q !F pE q ! F pE q ! of (3.2), indexed by p 2 Z, is a ltration of that complex if fF pE q gp2Z
is a ltration of E q for each q 2 Z . Any such ltration gives rise to
(3.4)
1
+1
a spectral sequence of bigraded modules in the following (completely
standard) manner: for each p; q 2 Z and each r 0 dene
Zrp;q := f a 2 F p E p
(3.5)
+q
check that dZrp r ;q r
dZ p r ;q r := 0, and set
(
1)
+(
1)
1
+1
+
1
: da 2 F p r E p
+
Zrp;q + F p E p
+1
+q
+q +1
g;
, where
2
1
(3.6) Erp;q := (Zrp;q + F p E p q )=(dZrp
+1
+
(r
1
1);q +(r
1)
1
+ F p E p q ):
+1
+
8
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
For xed r 0 the R-linear mapping d induces R-linear mappings
L
d : Erp;q ! Erp r;q r , and the direct sum Er := p;q Erp;q is thereby
endowed with the structure of a bigraded module with dierential dr
of bidegree (r; r + 1).
+
+1
Theorem 3.7. The sequence f(Er ; dr )gr is a spectral sequence.
0
For a proof see, e.g., [Mac, page 346].
Any R-linear mapping f : M ! N between R-modules can be
embedded into the nite complex
(3.8)
0!M
! N ! 0;
f
i.e., can be considered as one mapping within the cochain complex
(3.9)
! 0 ! 0 ,! E
!f E := N ! 0 ! 0 ! :
N admit ltrations fF pM gp2Z and fF pN gp2Z
0
:= M
1
0
When M and
and
f preserves these ltrations the spectral sequence construction immediately preceding Theorem 3.7 applies (assuming the trivial ltration
on 0). The resulting spectral sequence is the spectral sequence of the
linear (ltration preserving ) mapping f : M ! N .
The normal form algorithm considered in the next section is related
to the spectral sequence of the previous paragraph by taking M =
N = L to be a graded Lie algebra and by taking f := ad(`) for a xed
` 2 L. Unfortunately, the resulting spectral sequences do not admit
useful morphisms as one varies `. The construction in x6 will rectify
this problem.
We include the following identications so as to relate terms appearing in particular spectral sequence calculations to terms appearing in
normal form calculations. One has
(3.10)
and
Zrp;q = 0 and Erp;q = 0 if q 6= p; p + 1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
(3.11)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(a) Zrp;
(b) Zrp;
p
(c) Erp;
p
and
(3.13)
1
p+1
F M
F M \ f (F p r N ) + F p M
Fp M
p
F M \ f (F p r N )
; and
F p M \ f (F p r N )
F pN
f (Zrp r ; p r ) + F p N
F pN
f (F p r M \ f (F p N )) + F p N
F pN
f (F p r M ) \ F pN + F p N
+1
=
(d) Erp;
+
+1
=
p+1
=
p
1
+
+1
+1
1
+1
+
1
(
1)
+
+(
1)
+1
1
=
=
In particular,
(3.12)
= F pM \ f (F p r N );
= F pN;
Z p; p + F p M
= r p
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
8
>
>
>
>
>
>
>
>
<
1)
(
p
= F pM;
(b) Z p;
p+1
= F pN
(c) E p;
p
0
0
0
1
+1
1)
(a) Z p;
+1
F pM + F p M
Fp M
p
= F M=F p M
F pN
=
f (F p M ) \ F pN + F p N
= F pN=F p N :
+1
=
+1
+1
(d) E p;
p+1
0
+1
+1
+1
(a) Z p;
p
= F pM \ f (F p N );
(b) Z p;
p+1
= F pN
1
1
(c) E p;
>
>
>
>
>
>
>
>
:
(
p
1
(d) E p;
1
p+1
1
+1
F p M \ f (F p N )
F p M \ f (F p N )
F pN
=
:
f (F pM ) \ F pN + F p N
=
1
+1
+1
1
+1
+1
9
10
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
When there is an integer k such that the ltrations of the previous
paragraph satisfy F pM = F M = M and F pN = F k N = N for all
p < k one checks easily that for any such p and any r 0 one has
0
(3.14) p < k and r 0
8
>
>
>
>
>
>
>
<
)
>
>
>
>
>
>
>
:
Zrp;
Zrp;
p
Erp;
p
p+1
= M;
= N;
M +M
=
= 0 ; and
M
N
Erp; p =
= 0:
f (M ) + N
In particular, for k = 0 the spectral sequence of f : M ! N is then
+1
a 4th -quadrant spectral sequence. In the more general context of the
previous paragraph the spectral sequence is concentrated in the 2nd
and 4th -quadrants.
In practice the dierential dr : Erp; p ! Erp r; p r
is calculated
by means of elementary linear algebra: one computes the linear mapping f jF M \f 1 F + N = f jZ
in the top line of the following commutative diagram and interprets the results within the indicated quotients.
(3.15)
fj
\ 1( + )
p; p
p r
+
p
(
p
r
Zr
p;
r
)
F pM
=#
f
( + )+1
p
!
Fp
F
rN
+
N
#=
F p M \ f (F p r N )
F p rN
F M \ f (F N )
F p M \ f (F p r N )
F N
M ) \ F p rN + F p
1
p+r
+
#
+1
1
p+r
1
+
=#
Erp;
p
+
#
p+r
f (F p
+1
!
dr
+
#=
E
p+r;
r
(p+r )+1
+r +1
N
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
11
To ease notation express this last diagram as
Zrp;
(3.16)
p;r
!
f
p
#
Erp;
F p rN
+
#
p+r;r
p
! Erp
dr
and note that both p;r and p
lences in (3.11)).
+r;r
+r;
(p+r )+1
are epimorphisms (use the equiva-
Proposition 3.17.
(a) Choose e 2 Zrp;
and set [e] := p;r (e) 2 Erp; p. Then
the following statements are equivalent.
(i) [e] 2 ker dr ;
(ii) [e] survives to Er ;
(iii) f (e) 2 f (Zrp ; p ) + F p r N ;
(iv) f (e) 2 f (F p M ) \ F p r N + F p r N ; and
(v) there is an element a 2 Zrp ; p
such that f (e) f (a) =
p r
f (e a) 2 F
N.
Moreover, if a 2 Zrp ; p satises the condition in (v) then
e a represents the class of [e] in Erp; p.
p
F pM
+1
+1
( +1)
+ +1
1
+1
+
+ +1
+1
( +1)
1
+ +1
+1
( +1)
1
+1
(b) Suppose e^ 2 F p r N and set [^e] = p
+
+r;r
(^e) 2 Erp
+r;
(p+r )
. Then
the following statements are equivalent.
(i) [^e] 2 dr (Erp; p);
(ii) [^e] is killed by dr ; and
(iii) there is an element b 2 Zrp; p such that e := f (b) 2 F p r N
satises [^e] = [e] := p r;r (e).
+
+
Proof :
(a)
(i)
, (ii) : By denition.
, (iii) : By the commutativity of diagram
(i)
(3.16)
and the initial equality of (3.11d) (with p replaced by
p + r).
, (iv) : Use the nal equality of
(iii) , (v) : From the denitions.
(i)
(3.11d).
12
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
To prove the nal assertion rst note from F p M F pM that Zrp ; p Zrp; p F pM , hence a; e 2
F pM , and it follows from (v) that e a 2 Zrp; p. From
(e a) e = a 2 F p M we then see from the rst
equality in (3.11c) (with r replaced by r +1) that e a
represents the class of [e] in Erp; p.
+1
+1
( +1)
1
+1
+1
+1
(b)
(i)
(i)
,
,
(ii) : By denition.
(iii) : By the commutativity of (3.16).
q:e:d:
Example 3.18. Let N := TU (8; R ) denote the real graded Lie algebra
of Example 2.5(a), let M := F N , and dene f : M ! N by
1
f :m2M
(i)
where
7! ad(`)m = [`; m] 2 N ;
0
1
0 0 0 0 4 0 6 7
B0 0 1 0 0 0 0 12C
B
C
B0 0 0 1 0 3 8 0 C
B
C
B0 0 0 0 1 0 0 0 C
` = B0 0 0 0 0 1 0 0 C :
(ii)
B
C
B
C
B0 0 0 0 0 0 0 2 C
@0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
(In fact f : M ! M : we write f : M ! N so as to conform with the
notation used thus far in the section.) Assuming the induced grading
on M , i.e., M := 0 and Mp = Np for p 1, the mapping easily seen
to satisfy the hypotheses surrounding (3.8) and (3.14); we compute the
associated spectral sequence. In the notation of (3.14) we have k = 0,
and that sequence is therefore a 4th -quadrant spectral sequence. In
particular, we only need compute Erp; p and Erp; p for p 0 and
r 0.
Throughout the calculations we let epk 2 M denote the 8 8 matrix
in ltration p with (k; k +p)-entry 1 and all other entries 0, 1 p 7
and 1 k 8 p. Note that (ep ; : : : ; ep; p) provides a(n ordered)
basis of Lp. Equivalence classes (cosets) of the epk will be denoted
[epk ], regardless of the particular factor space.
0
+1
1
8
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
13
The E0 Terms : We have E p; p = F p M=F p M ' Mp and E p;
F pN=F p N ' Np for all p 0 by (c) and (d) of (3.12).
The E1 Terms : From ` 2 M = F N we have
+1
0
p+1
0
=
+1
1
f (F p M ) F p N ;
(iii)
+1
whereupon from (c) and (d) of (3.13) we conclude that E p; p =
F pM=F p M = E p; p and E p; p = F pN=F p N = E p; p . These
isomorphisms would generally be indicated by writing E = E (or
E = E ).
The diagrams for both the E and E terms both begin with that
shown below, wherein the notation Eipq for i = 0; 1 is replaced by
nR := R R to indicate a basis dependent vector space isomorphism E pq ' R n and no label is associated with trivial spaces. The
bases are always induced from the given basis (epj ) of M N , e.g.,
the basis for Ei ; ' M for i = 0 and 1 is ([e ]; : : : ; [e ]). The
distinction between the two diagrams becomes evident only when the
dierentials are added to complete the pictures: for E the dierential
would be indicated by vertical arrows between nR and nR , and for
E by horizontal arrows from nR to (n 1)R .
1
+1
+1
0
+1
+1
1
0
0
0
1
1
0
1
1
1
1
11
17
0
1
0
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
@7 R
@@ @@
@ 7 R -@@6 R
@@ @
@@6 R -@@5 R
@@ 5 R@@ 4 R
@ - @@
@@ 4 R -@ 3 R
@@ @@
@@3 R -@@2 R
@@ 2 R@-@ 1 R
@@ @@
@ @
@
r
r
r
r
r
r
r
r
r
r
r
r
r
The E and E ; d terms.
0
1
1
r
14
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
The E2 Terms : This requires calculating the mappings d : E p; p !
E p ; p, and we do so as in (3.16) (more precisely, as in (3.15))
with r = 1. The condition ` = 0 gives Z p; p = F p N = F pM
for p 1, and as a consequence it suÆces to calculate the eect of
ad(`)jF N : F pN ! F p N on the basis elements epj and then pass
1
1
+1
1
0
1
+1
p
to quotients. The calculations are completely straightforward, and the
results are summarized in the following table, wherein the initial entry
p = 1; [e ] 7! [e ] indicates that d : [e ] 2 E ; 7! [e ] 2
E ; , etc.
1
11
2
21
1
11
1
21
1
1
1
[e
[e
[e
[e
[e
[e
[e
11
12
13
p=1
14
15
16
17
!
[e ]
!
[e ]
! [e ] [e
! [e ] [e
! [e ]
! [e ]
!
0
]
]
]
]
]
]
]
[e
[e
[e
[e
[e
[e
21
21
22
22
23
23
24
]
]
22
p=2
23
24
24
25
25
[e
[e
[e
[e
[e
31
32
p=3
33
34
35
]
]
]
]
]
! [e ]
! [e ]
! [e ]
! [e ]
! [e ]
41
42
p=4
42
43
31
32
32
! 0
! 0
! [e ]
51
52
53
p=6
33
]
33
34
35
[e
[e
[e
[e
41
42
43
44
! [e ]
! 0
! [e ]
! [e ]
]
]
]
]
51
52
53
44
[e ]
[e ]
[e ]
p=5
26
!
[e ]
!
[e ]
! [e ] [e
! [e ]
! [e ]
! [e ]
]
]
]
]
]
]
[e ]
[e ]
61
62
62
!
!
0
0
We can use these calculations to illustrate the spectral sequence
jargon introduced earlier: [e ] and [e ] + [e ] + [e ] + [e ] 2 E
survive to E ; [e ] 2 E does not, and [e ] 2 E ; is killed (by
[e ] 2 E ; ), as is [e ] (by [e ]). In particular, [e ] and [e ]
must survive to E and represent 0 .
17
12
13
14
3
2
2
21
11
1
31
15
1
2
1
2
42
1
2
32
31
42
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
15
From the calculations above the cohomology E of E , described in
terms of associated generators (i.e., basis elements), is easily seen to be
2
E
E
E
E
E
E
E
1;
1
2
2;
1
2
2;
2
2
3;
3
2
4;
4
2
5;
5
2
6;
5
2
:
:
:
:
:
:
:
[e ] a = [e + e + e + e ]
[e ]
b = [e + e + e ]
;
c := [e + e ]
[e ]
[e ]
[e ]
[e ]
17
12
13
14
15
26
22
23
24
32
33
42
51
52
61
and the associated diagram for E is therefore
0
1
2
3
4
5
6
@7r R
0 @r
2
1
2
3
4
5
6
7
1
7
8
@@ @@
@H2 R @ 1 R
@@HH @@
@H
1H
RH
j@ 0
H
@HH @
@@H1 HR@Hj@ 0
@HHH @
@@H1 HR@@ 0
Hj
@HHH @
@@H2 HR@@ 1 R
@HHHjH @
@@H2 HR@Hj@ 1 R
@ @
@@ @@
r
r
r
r
r
r
r
r
r
r
r
r
The E ; d term.
2
2
wherein 0 denotes the trivial module. (Recall that unlabeled vertices
also represent the trivial module.)
The E3 Terms : We need to compute d : E p;
2
2
p
! Ep
+2;
p
2
1
: From
the last diagram we see that only possible nontrivial components of
this homomorphism arise in the contexts E ; R ! E ; R and
E ; 2R ! E ; R .
Applying (3.16) with r = 2 we obtain the following analogue of the
rst collection of displayed formulas within the discussion of the E
4
2
5
2
5
7
4
6
5
2
6
2
2
16
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
terms:
[e ] ! 2[e ]
[e ] !
0
The diagram for the E terms appearing below is an easy consequence.
p = 4 [e ] ! 2[e ]
42
p=5
62
51
71
52
3
0
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
@@ @7@R
@ @
@@2 R @@1 R
@ @
@@1 R @@0
@ @
@@Q1 R @@0
@Q@QQ1 R@@ 0
@@QQQ@@
@ 1 QRQs@ 1 R
@@ @@
@@2 R @@0
@@ @@
r
r
r
r
r
r
r
r
r
r
r
r
r
r
The E ; d term.
3
3
This seems an appropriate place to ease the formality of our presentation: in practice the observations resulting in the E diagram would
more likely be stated along the following lines.
The space E ; is generated by [e ], which is mapped
by d to 2[e ] = 0 2 E ; ([e ] was killed by [e ]).
The mapping d : E ; ! E ; is therefore the zero
transformation, and as a consequence [e ] survives to
E . The class [e ] is represented in E by [e + 2e ].
3
4
4
42
2
6
2
62
5
62
2
4
2
4
6
2
53
5
2
42
3
42
3
42
53
The space E ; is generated by [e ] and [e ], and one
checks that d ([e ]) = 2[e ] and d ([e ]) = 0.
The E4 Terms : The only possible nontrivial (component of) d :
E p; p ! E p ; p is (the restriction to) E ; ! E ; . However, one
checks that E ; is generated by [e + e ], and that d carries this
class to 0. E = E follows.
5
5
51
2
2
51
71
52
2
52
3
+3
3
+2
3
3
3
3
6
5
3
3
32
3
4
3
3
33
3
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
17
The E5 Terms : The only possible nontrivial d is E ; ! E ; .
The rst of these spaces in generated by [b], and d ([b]) = 0. E = E
2
4
2
6
4
5
4
4
5
4
follows.
The E6 Terms : The only possible nontrivial d is E ; ! E ; .
The rst of these spaces in generated by [e ] and [a], and d annihilates both. E = E follows.
The E7 Terms : The dierential d is trivial, hence E = E .
The E1 Terms : All dr with r 6 and trivial, hence E1 = E =
p;q
E (in the sense that E1
= E p;q for all (p; q ) 2 Z Z). There is a
single generator ! for E1; and a single generator ! for E1; ; all
p; p
the other vector spaces E1
are trivial.
We have calculated the spectral sequence of f = ad(`) : M ! N
1
5
17
6
1
6
5
5
5
5
6
7
6
6
3
3
2
2
1
6
6
5
+1
completely, and in the process have established strong convergence.
5
18
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
A Brief Introduction to Normal Form Theory
L
Throughout this section L = L denotes a Z-graded R-Lie algebra with L = 0 if s < 0 and : L ! L is used to denote
the associated projections. We write the typical element of L as `
and view each L as a subspace of L by means of the obvious section L ! L, i.e., we identify an element ` 2 L with the element
+ 0 + ` + 0 + 2 L when confusion cannot otherwise result.
Suppression of notational reference to the sections L ! L is a com4.
s
s
s
s
s
s
s
s
s
s
s
s
mon abuse of notation when dealing with normal forms, but can lead
to problems when spectral sequences enter the picture.
For the entire section we x an element ` 2 L . We do not exclude
the choice ` = 0.
0
0
0
Denition 4.1. An element ` = ` + ` + + `s + 2 L is in
classical normal form to order s 0 if `j 2 ker(ad(` )jL ) for j =
0; : : : ; s, and is in classical normal form if this is the case for all s 0.
0
1
1
0
j
In other words, ` is in normal form (to order s) if [` ; `j ] = 0 for
all 0 j ( s). Note from [` ; ` ] = 0 that ` is always in classical
normal form to order 0.
An element ` 2 L splits L if
0
0
0
0
0
Lj = ker(ad(` )jL ) im(ad(` )jL );
(4.2)
0
0
j
j
j 1:
Proposition 4.3. (The Classical Normal Form Algorithm) Suppose ` splits L and ` = ` + + `s + is in classical normal
form to order s. Write `s = `Ks + `Is in accordance with the
decomposition (4.2) with j = s + 1. Choose m 2 Ls such that
ad(` )m = [` ; m] = `Is . Then ` + ad(m)` is in classical normal form
to order s + 1 and Js (` + ad(m)`) = Js (`).
0
0
+1
+1
+1
+1
0
0
+1
This formulation is adapted from [CKR], but did not originate therein.
Proof : This is evident from the following calculation, where in each
line the nal dots represent terms in
Q
t s+2
Lt .
The \classical" designation is not standard: it has been added to distinguish
these normal forms from those introduced later.
1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
` + ad(m)` = Js (`) + `Ks
= Js (`) + `K
s
= Js (`) + `K
s
= Js (`) + `K
s
= Js (`) + `K
s
+1
+1
+1
+1
+1
+ `Is + + [m; ` + + `s
+ `Is + [m; ` ] + + `Is
[` ; m] + I
+ `s
`Is + + :
0
+1
+1
19
+ ]
0
+1
+1
0
+1
+1
q:e:d:
For m 2 L dene ad (m) := idL : L ! L, and if i 1 and
adi (m) : L ! L has been dened set adi (m) := ad(m) Æ adi (m) :
L ! L.
To see how the algorithm can be applied in practice assume, for the
remainder of the section, that R is a eld of characteristic 0. Then
for any m 2 F L a linear mapping expad : L ! L is dened by
1
X
1 i
(4.4)
ad (m):
expad(m) :=
i
!
i
0
1
1
1
=0
Indeed, by (2.4) and the assumption m 2 F
(4.5)
1
L
the formal expression
expad(m)` = ` + [m; `] + [m; [m; `]] + 1
2
involves only nite sums in each Lp , and therefore represents a welldened element of L. In fact expad(m) : L ! L is a K -Lie algebra
automorphism , i.e.,
(4.6)
expad(m)[`; `^] = [expad(`); expad(`^)];
m 2 F L;
`; `^ 2 L :
2
1
Example 4.7. Fix an integer n 1 and let L := TU (n; R) denote
the graded Lie subalgebra of gl(n; R) introduced in Example 2.5(a).
Choose M 2 F L and B 2 L. Then one sees by writing out the
Taylor series for f (t) = eM t Be M t at t = 0 and evaluating at t = 1
1
that
(i)
expad(M )B = eM Be
M
:
The next proposition shows that the adjoint representation in algorithm (4.3) may be replaced with expad.
2
When dimK L < 1 this is standard; for the general case see, e.g., [Se] or [BC].
20
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Proposition 4.8. Suppose ` splits L and ` = ` + + `s + is in classical normal form to order s. Write `s = `Ks + `Is in
accordance with the decomposition (4.2) with j = s + 1. Choose m 2
Ls such that ad(` )m = [` ; m] = `Is . Then expad(m)` is in
classical normal form to order s + 1 and Js (expad(m)`) = Js (`).
0
0
+1
+1
0
0
+1
+1
+1
Proof : Immediate from Proposition 4.3 and (4.5).
q.e.d.
Remark 4.9. The advantage of Proposition 4.8 over Proposition 4.3
is suggested by Example 2.5(a), where successive applications of the
normal form algorithm to a given A 2 T are now seen to produce
a collection of (generally non-unique) matrices Mn ; Mn ; : : : ; M 2
F T such that conjugating A by the product eM eM1 converts A
to the appropriate classical normal form.
1
1
1
n
Group actions enter the picture by rst noting that the graded vector
subspace G := F L L is a ltered group w.r.t. the binary operation
dened by the Campbell-Hausdor formula
1
(4.10)
m n = m + n + [m; n] + [m; [m; n]] + 1
1
2
12
(e.g., see [BC] and/or [Se, 14.15]) : the ltration fF p Ggp of G is
Q
dened by the inherited grading, i.e., F pG := q Gq , where Gq := Lq
for all q 1; the identity element is 0; the inverse of m 2 G is m.
Denition (4.10) is designed so as achieve
1
1
(4.11)
expad(m n) = expad(m) expad(n);
m; n 2 G ;
where expad(m) expad(n) := expad(m) Æ expad(n), and it follows that
the mapping (m; `) 2 G L ! expad(m)` denes a left action of G
on L by K -Lie algebra automorphisms (recall (4.6)). One can now
interpret successive applications of Proposition 4.8 as the iterated construction of an orbit representative of `, although for the actual existence proof one needs to establish convergence in the ltration topology
of G .
There are two signicant problems with the classical theory:
classical normal forms obtained by successive applications of Proposition 4.8 are generally not unique; and
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
21
when ` 2 L
does not split L there is no algorithm to guarantee
that one can always convert an element ` = ` + ` + 2 L to
classical normal form.
0
0
0
1
The rst problem was generally treated by attempting \further renements" of elements in classical normal form; the second by replacing
ker(ad(` )jL ) in (4.2) with a suitable complement of im(ad(` )jL )
(often associated with the representation theory of sl(2; C )). Of course
each of these approaches required modications of Denition 4.1. A.
Baider [B] gave an elegant solution to both problems by replacing
im(ad(` )jL ) in the decomposition of Lj with a generally larger subspace and assuming a prescribed complement, e.g., the orthogonal complement w.r.t. a given inner product on Lj .
To describe Baider's method assume ` = ` + ` + 2 L has been
given, dene
(4.12)
Cs (`) := f m 2 G := F L = F G : [m; `] 2 F s L g;
s 0;
0
0
0
j
j
j
0
1
3
1
1
1
+1
and then dene
(4.13)
Vs (`) := s
ad(`)(Cs (`))
1
+1
1
+1
Ls ;
+1
s 0:
Note that when s 0 and m 2 Cs (`) one has
1
(4.14)
[expad(m)`] = [` + [m; `]] 2 L=F s L ;
+1
this is all one needs to mimic the classical normal form algorithm.
Continuing with the notation of the previous paragraph assume that
for each s 1 a complement Ys(`) Ls of Vs (`) has been chosen
which depends only on Js (`), hence that
1
1
(4.15)
Ls = Ys(`) Vs (`);
s 1:
1
In particular,
(4.16)
Vs (`) = Ls
,
1
Ys(`) = 0:
To involve all non-negative indices in the denition of Vs (`) dene
1
(4.17)
Y (`) := L :
0
0
Baider refers to the Lie subalgebra Cs1 (`) L as the s-order \centralizer" of
`, and employs slightly dierent notation. Our notation is designed to make the
connection with spectral sequences more transparent.
3
22
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
A choice of complements as in (4.15) is called a splitting convention
in [CK] and a style in [Mur1, Mur2].
Denition 4.18. An element ` = ` + ` + 2 L is in normal form
to order s 0 (w.r.t. the assumed splitting convention) if `j 2 Yj (`)
for j = 0; : : : ; s, and is in normal form if it is in normal form to order
s for all s 0.
0
1
Examples for any splitting convention: any ` 2 L is in normal form
to order 0 ; 0 2 L is in normal form.
Proposition 4.19. Suppose ` = ` + + `s + is in normal form
to order s 0. Write `s = `Ys + `Vs in accordance with the
decomposition (4.15) (with s replaced by s + 1). Choose m 2 Cs (`)
such that s ad(`)m = `Vs . Then expad(m)` is in normal form
to order s + 1 and Js (expad(m)`) = Js (`).
0
+1
+1
+1
1
+1
+1
Proof : Immediate from Proposition 4.3, (4.14), and the assumption
that Ys (`) depends only on Js(`).
q.e.d.
+1
We can now be more explicit about one of the goals of the paper:
we will show, in somewhat greater generality, that the calculations involved in applying Proposition 4.19 to specic normal form problems
are simply special cases of spectral sequence calculations as in Example 3.18. However, since the present section is intended to introduce
normal forms as treated by practitioners, our discussion of the actual
connections with spectral sequences is postponed to a later section (see
x6).
Baider's main result, which we state without proof, is as follows.
Theorem 4.20. (A. Baider [B]) The G -orbit of any element ` 2
L contains a unique element `N in normal form, and if the normal
form algorithm dened by Proposition 4.19 is used to produce elements
ms 2 G to convert expad(ms m )` to normal form of order
s + 1 the sequence fms m g converges in G to an element m
such that expad(m)` = `N .
1
1
1
Baider refers to these unique normal forms as special forms [B], and
the terminology hypernormal forms is also encountered [Mur1, Mur2].
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
23
The calculation of the subspaces C (`) L and V (`) L is
always straightforward. Specically, one sees from the denition that
C (`) = F L = G , and from (2.4) that V (`) := ad(`)(C (`)) =
ad(`)(G ) = ad(` + )(L F G ) = ad(` )(L )
= ad(` )(L ), where in writing ad(` )(L ) we are identifying
ad(` )(L ) with its image in L under the obvious section L ! L
of . In summary:
1
0
1
1
1
0
1
1
0
1
0
0
0
0
2
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
C (`) = G
(4.21)
V (`) = ad(` )(L ):
and
1
0
1
0
1
1
In special cases the calculation of Vs (`) is also easy: for any s 0
one has
1
+1
Ls Cs (`)
(4.22)
1
(more precisely:
(4.23)
Ls = s(Cs (`))), hence
ad(` )(Ls ) Vs (`) ;
1
1
0
and it follows that
(4.24)
ad(` )(Ls ) = Ls
0
+1
+1
+1
) Vs (`) = Ls
and Ys (`) = 0
1
+1
+1
+1
+1
(recall (4.16)).
Other easy cases arise. For example, when ` = 0 one sees from
(4.12) that C (`) = G , whence from (2.4) that V (`) = ad(`)(G )
= ad(` )L , i.e.,
0
1
1
1
1
2
2
1
` =0
(4.25)
0
)
V (`) = ad(` )L :
1
1
2
1
Unfortunately, the determination of Cs (`) and (thence) Vs (`) can
in general be a daunting task, although it is diÆcult to appreciate this
assertion until one begins working with specic examples. (With the
spectral sequence approach the calculation of Vs (`) becomes completely systematic, albeit tedious at times.) On the other hand, as will
be seen in Example 4.34, when utilizing the normal form algorithm one
can sometimes verify that `s 2 Vs (`) without complete knowledge
of either Cs (`) or Vs (`), in which case it is clear from the normal
form algorithm that the normal form `N must satisfy `Ns = 0.
An obvious approach to computing Cs (`) is to work upward through
the ltration
1
1
+1
1
+1
+1
1
+1
1
1
+1
+1
1
(4.26)
C s(`) C s (`) Cs (`)
1
1
2
1
24
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
of Lie subalgebras dened by
(4.27)
Csp p (`) := f m 2 F pG : ad(`)m 2 F s
+1
+1
L g;
p = s; s 1; : : : ; 1;
and with this in mind we dene the initial terms Isp
by
(4.28)
Isp
p+1
(`) = p Csp
p+1
(`)
Lp;
p+1
(`) of Csp
p+1
(`)
p = s; s 1; : : : ; 1:
The initial terms of C s (`) are easy to compute: we claim that
1
I s(`) = ker ad(` )jL :
Indeed, for m = ms + ms + 2 F sL see from (2.4) that
ad(`)m = [` + ` + ; ms + ms + ]
= [` ; ms ]s + fterms in F s Lg;
(4.29)
0
1
s
+1
0
1
+1
+1
0
and the claim follows. As a consequence of (4.29) and (4.26) we see
that
(4.30)
` = 0 and s 1
)
0
I s (`) = s C s (`) = Ls :
1
1
However, from the denitions (and the subspace identication conventions) one sees that I s(`) C s (`), and it follows that
(4.31)
` = 0 and s 1 ) Ls C s(`) and ad(` )Ls Vs (`):
1
1
1
0
1
1
+1
We need a practical characterization of the initial terms of Csp p (`).
Suppose 1 p < s and mp 2 Lp. Then mp completes in Csp p (`)
if there is an element m
^ 2 F p G such that mp + m
^ 2 Csp p (`).
+1
+1
+1
+1
Proposition 4.32. For any 1 p < s and any mp 2 Lp the following statements are equivalent:
(a) mp 2 Isp p (`), i.e., mp is an initial term of Csp p (`);
(b) the element mp completes in Csp p (`);
(c) one has
+1
+1
+1
ad(`)(mp) 2 ad(`)(F p
+1
L) + F s L ;
+1
and
(d) one has
[0] = [ad(`)(mp)] 2 F p L= ad(`)(F p
+1
L) + F s L :
+1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
Proof : For m
^ 2 Fp
+1
mp + m
^ 2 Csp
p+1
L
(`)
25
we have
,
,
ad(`)(mp + m
^ ) = 0 mod F s L
ad(`)mp = ad(`)( m
^ ) mod F s
+1
+1
L;
q.e.d.
and the equivalences follow.
For normal form calculations the equivalence (a) , (c) is the most
important, and for ease of reference we record this separately: for mp 2
Lp we have
(4.33) mp 2 Isp
p+1
,
(`)
ad(`)(mp) 2 ad(`)(F p
+1
L) + F s L :
+1
Example 4.34. We oer a concrete normal form calculation within
the real graded Lie algebra L = TU (8; R ) (see Example 2.5(a)). Nilpo-
tent cases often present problems in normal form calculations (in part
because ` does not split L), and we have therefore chosen to consider such an example in some detail. The choice n = 8 allows us
to illustrate all the important concepts while keeping the calculations
(which were done with MAPLE) within reason. The presentation is
designed to emphasize the underlying systematic procedure, and as
a result is more formal than necessary for such an elementary example. The splitting convention is that dened by the inner product
hA; B i := tr(A B ) on L, i.e., in the direct sum decompositions (4.15)
we take Yp (`) := Vp (`)? Lp .
We compute the normal form of the nilpotent matrix
0
1
0
(i)
0
B0
B
B0
B
0
` := B
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
4
0
0
1
0
0
0
0
0
0
3
0
1
0
0
0
6
0
8
0
0
0
0
0
1
7
12C
C
0C
0C
C
0C
C
2C
C
0A
0
appearing (not coincidentally) in Example 3.18, and to use the methods
introduced we write ` in the form ` + ` + + ` , wherein ` denotes
0
1
7
0
26
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
the zero matrix,
0
0 0 0 0
B0 0 1 0
B
B0 0 0 1
B
0 0 0 0
` =B
B
B0 0 0 0
B
B0 0 0 0
@0 0 0 0
0 0 0 0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0 0 0
0
B
C
0 0 0
0
B
C
0C
B0 0 0
B
0C
B0 0 0
C
C; ;` = B
0C
B0 0 0
B
0C
B0 0 0
C
@0 0 0
A
0
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
7
0C
C
0C
0C
C
:
0C
C
0C
C
0A
0
The normal form of ` to order s 0 is written ` s = ` s + ` s +
` s + .
Order 0 : As noted immediately following Denition 4.18, the element
` is automatically in normal form to order 0, hence ` = `.
Order 1 : Since ` = ` = 0 we see from (4.21) that V (` ) = 0,
hence Y (` ) = L , and we conclude that ` is also in normal form
to order 1. It follows from the uniqueness of normal forms that ` =
` = `. In the notation of Remark 4.9 we take M to be the zero
matrix, and eM1 is then the identity matrix I = I .
Order 2 : By (4.25) we have V (` ) = ad(` )(L ) = ad(` )(L ),
and by elementary calculation one veries that this last subspace of
L consists of those elements mij 2 L with m = 0. From the
denition Y (` ) = V (` )? we conclude that V (` ) consists of
those elements mij 2 L in which all entries other than m must be
zero, hence ` 2 Y (` ), and ` = ` = ` = ` follows. We take
M as the zero matrix, resulting in eM2 = I .
Order 3 : Check that the matrix M 2 L with 3 in the (4; 6)
position and zeros elsewhere satises ad(` )(M ) = ` . It follows
from (4.31) that ` 2 V (` ), hence that ` = 0. To calculate
` completely note that eM2 = I + M ; then check that
( )
( )
( )
0
1
( )
2
(0)
(0)
1
(0)
1
0
0
(0)
1
(0)
1
(1)
(0)
1
8
1
(1)
(1)
2
1
1
2
2
(1)
2
1
1
68
(1)
1
2
(1)
2
2
(1)
2
2
1
68
1
(2)
(1)
(0)
2
2
2
(2)
(2)
(2)
3
1
3
2
3
(3)
(2)
3
(3)
2
0
` = expad(M )` = eM ` e
(3)
(2)
2
2
(2)
M2
0
B0
B
B0
B
0
=B
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
4
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
6
0
8
0
0
0
0
0
1
7
12C
C
0C
6C
C
:
0C
C
2C
C
0A
0
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
27
Order 4 : We proceed as in the Order 3 case after noting with the aid
of (4.31) that for any 2 R the matrix
0
0
B0
B
B0
B
0
M =B
B
B0
B
B0
@0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0 0
0
0 0 0
0 0
0 0
0 0
0 0
1
0
0
0
8
0
0
0
0
0
0C
C
0C
0C
C
2L
6C
C
0C
C
0A
0
satises ad(` )(M ) = ` , hence `
(3)
eM
3
3
0
(3)
(4)
4
4
1
B0
B
B0
B
0
=B
B
B0
B
B0
@0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
from which one obtains
4
0
0
1
0
0
0
0
C (`
3
3
(3)
1
2
(3)
1
2
(3)
3
= 0. One has
1
0 0
0
0 0 0
1 0
0 1
0 0
0 0
16 0
0 3C
C
0
0C
8
0C
C
;
0
6C
C
0
0C
C
1
0A
0
1
0
0 0 0 0 0
B0 0 1 0 0
B
B0 0 0 1 0
B
0 0 0 0 1
` = expad(M )` = B
B
B0 0 0 0 0
B
B0 0 0 0 0
@0 0 0 0 0
0 0 0 0 0
This illustrates non-uniqueness within the Mj .
(4)
) C (` ) C (` )
(3)
4
0
0
0
0
1
0
0
0
6
0
0
0
0
0
0
0
1
17
12 C
C
2 C
0 C
C
:
0 C
C
2 C
C
0 A
0
Order 5 : We make the choice = 0 in the previous step; the matrix
` is then seen to be in normal form to order 5, hence ` = ` .
(By the uniqueness of normal forms any other choice for would have
[ultimately] resulted in an ` with the same 5-jet.) We take M = 0,
hence eM = I .
Order 6 : Here the method used for Orders 3 and 4 fails: one easily
veries that ` 2= ad(` )(L ), and as a result we cannot appeal to
(4.31) to conclude that ` 2 V (` ). This is the rst case in which
(4)
(5)
(5)
(4)
4
4
(6)
(5)
5
(5)
1
(5)
6
5
Proportion 4.32, in the guise of (4.33), plays a signicant role. We ex
amine the initial terms I p p (` ) := p C p p (` ) as p decreases
(5)
(5)
5
+1
5
+1
28
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
from 4, recalling from (4.33) that
(ii)
mp 2 I p
5
p+1
,
(` )
(5)
ad(` )mp 2 ad(` )(F p
(5)
(5)
+1
L) + F L :
6
We oer a somewhat detailed presentation of this case so as to emphasize the completely elementary nature of the calculations.
The Initial Terms I24 (`(5)) = 4 C24 (`(5))
: In this case (ii)
becomes
(iii)
m
4
2 I (`
4
(5)
2
)
,
ad(` )m
(5)
2 ad(`
(5)
4
)(F
L) + F L :
5
6
However, from ` = ` = 0 and (2.4) we see that ad(` )(F
F L, whereupon (iii) reduces to
(5)
(5)
0
0
5
L) 6
2 I p(` ) , ad(` )m 2 F L :
The Lie subalgebra F L L consists of all matrices of the form
1
0
0 0 0 0 0 0 B0 0 0 0 0 0 0 C
C
B
m
(iv)
(5)
4
(5)
6
4
2
6
B0
B
B0
B
B0
B
B0
@0
0
0
0
0
0
0 0
(v)
and for a typical element
0
0 0 0
B0 0 0
B
B0 0 0
B
0 0 0
(vi)
m := B
B
B0 0 0
B
B0 0 0
@0 0 0
0 0 0
4
we have
(vii)
0
0
B0
B
B0
B
0
ad(` )(m ) = B
B
B0
B
B0
@0
0
(5)
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 m
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
m
26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0C
0C
C
;
0C
C
0C
C
0A
0
1
0
0
m
37
0
0
0
0
0
m
48
0
0
0
0
m
0
0
0
0
0
0
0
0
0 C
C
0 C
15
C
C
C
C
C
C
A
0
m
37
0
0
0
0
0
0
2L
4
1
0
2m C
26
m
48
0
0
0
0
0
C
C
C
C
C:
C
C
C
A
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
29
It follows immediately from (vii) that I (` ) = C (` ) consists
of those m as in (vi) with m = m = m = 0, i.e., that a matrix
m 2 L completes in C (` ) if and only if this matrix has the form
4
(5)
4
4
15
4
4
37
4
4
2
(5)
2
48
(5)
2
0
0
B0
B
B0
B
0
m := B
B
B0
B
B0
@0
0
(viii)
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 m
0 0
0 0
0 0
0 0
0 0
0 0
1
0
0
0
0
0
0
0
0
26
0
0C
C
0C
0C
C
:
0C
C
0C
C
0A
0
Now let m = m + m
^ 2 I (` ) + F L be an arbitrary element of
C (` ). Then ad(` )m
^ 2 F L, and we conclude from (vii) that
ad(` )(C (` )) has the form seen in (v) when the (1; 7)-entry has
been replaced by 0. Since the (1; 7)-entry 6 of ` (= ` ) is not zero
this means that more work is required to determine if ` 2 V (` ).
We therefore ascend to C (` ).
4
4
4
(5)
6
2
(5)
(5)
7
2
(5)
4
(5)
2
(5)
(4)
(5)
1
(5)
6
3
(5)
3
The Initial Terms I33(`(5)) = 3 C33 (`(5)) : Here (ii) becomes
(ix)
m
2 I (`
3
3
(5)
3
,
)
One checks that ad(` )F
duced to
m
(5)
3
4
2 I (`
(5)
3
2 ad(` )F ` + F `:
(5)
L + F L = F L,
(5)
3
3
ad(` )m
6
)
,
5
ad(` )m
(5)
4
6
and (ix) is thereby re-
2 F L:
5
3
Now check that for the typical element
0
0
B0
B
B0
B
0
m := B
B
B0
B
B0
@0
0
3
0
0
0
0
0
0
0
0
0 m
0 0
0 0
0 0
0 0
0 0
0 0
0 0
14
0
m
25
0
0
0
0
0
0
0
0
m
36
0
0
0
0
0
0
0
0
m
47
0
0
0
0
0
0
0
0
1
C
C
C
C
C
m58 C
C
0 C
C
0 A
0
2L
3
30
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
we have
0
0
B0
B
B0
B
0
ad(` )m := B
B
B0
B
B0
@0
0
(5)
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
m
0
0
0
0
0
0
0
0
and as a result we see that I
the form
0
0 0
B0 0
B
B0 0
B
0 0
m := B
B
B0 0
B
B0 0
@0 0
0 0
3
3
0
0
0
0
0
0
0
14
m
0
36
m
0
0
0
0
0
0
25
1
0
0
m
0
0 C
C
2m C
47
m
0
0
0
0
0
36
58
0
0
0
0
(` ) consists of those matrices of
(5)
0
0
0
0
0
0
0
0
3
0 0
0 m
0 0
0 0
0 0
0 0
0 0
0 0
0
0
25
m
0
0
0
0
0
0
0
0
25
0
0
0
0
0
C
C
C;
C
C
C
A
L
3
of
1
0
0C
C
0C
0C
C
;
0C
C
C
0C
0A
0
and the typical element of C (` ) has the form
3
(5)
3
0
0
B0
B
B0
B
0
m +m
^ =B
B
B0
B
B0
@0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 m
0 m
0 0
0 0
0 0
0 0
0 0
0 0
15
25
m
m
m
m
m
m
16
26
25
0
0
0
0
0
18
C
C
38 C
C
48 C
:
0 C
C
0 C
C
0 A
27
28
37
0
0
0
0
0
0
By direct calculation one checks that
0
0
B0
B
B0
B
0
ad(` )(m + m
^) = B
B
B0
B
B0
@0
0
(5)
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
m
m
m
m
17
1
C
C
0 C
C
0
0
0
0
0
0
0
0
0
0
0
0
0
0C
;
0C
C
0C
C
0A
0
and we immediately conclude, as in the nal assertion of the previous
case, that additional work is needed to determine if ` 2 V (` ).
(5)
6
1
5
(5)
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
31
The remaining initial terms relating to the order 6 calculation, i.e.,
I (` ) and I (` ), are handled analogously, and in both cases one
nds that the typical matrices in ad(` )(C j j (` )) again have 0
as the (1; 7)-entry, j = 2; 1. However, since these remaining terms
exhaust all possibilities we are now able to conclude that V (` )
consists of those matrices as in (ii) with the upper-right entry replaced
Y
V1
by 0. The splitting ` = ` + ` 6 of Proposition 4.19 is therefore
given by
2
(5)
1
4
(5)
5
(5)
(5)
6
1
(5)
6
(5)
(5)
6
0
0
B0
B
B0
B
B0
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(5)
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
1
0
0
0
0C B0
C B
0C B0
B
0C
C B0
C+B
0C B0
B
0C
C B0
A
@0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
12C
C
0C
0C
C
;
0C
C
0C
C
0A
0
and from (iv) we see that the matrix
0
0
B0
B
B0
B
0
M := B
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
satises ad(` )M = `
(5)
(5)
5
0
0
0
0
0
0
0
0
V61
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
(6)
5
(5)
1
0
0C
C
0C
0C
C
2 I (` ) C (` )
0C
C
0C
C
0A
0
4
(5)
4
2
(5)
2
. One has expad(M ) = I + M , hence
5
0
` = eM ` e
0
0
0
0
0
0
0
0
M5
0
B0
B
B0
B
0
=B
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
5
0
0
0
0
1
0
0
0
6
0
0
0
0
0
0
0
1
17
0 C
C
0 C
0 C
C
:
0 C
C
2 C
C
0 A
0
32
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Order 7 : The calculation of `
note that for
0
0
B0
B
B0
B
0
M := B
B
B0
B
B0
@0
0
one has eM6 = I + M
6
6
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
and
0
0
0
0
0
0
0
0
(7)
involves no new ideas: suÆce it to
0 17=2
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
1
0
0C
C
0C
0C
C
2 I (` )
0C
C
0C
C
0A
0
5
(6)
2
1
0 0 0 0 0 0 6 0
B0 0 1 0 0 0 0 0C
C
B
B0 0 0 1 0 0 0 0C
B
0 0 0 0 1 0 0 0C
C
` = eM6 ` e M6 = B
C:
B
B0 0 0 0 0 1 0 0C
C
B
B0 0 0 0 0 0 0 2C
@0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
This is the unique normal form of the matrix ` given in (i), and from
Theorem 4.20 we see that a matrix which conjugates ` to this normal
form is given by
1
0
1 0 0 4 0 7=2 16 0
B0 1 0 0 0
6
0 0C
C
B
0
0 0C
B0 0 1 0 0
B
3
8 0C
C
B0 0 0 1 0
M7 M6
M1
e e e = B0 0 0 0 1 0
C:
0
6
C
B
B
1
0 0C
C
B0 0 0 0 0
@0 0 0 0 0
0
1 0A
0 0 0 0 0 0
0 1
(7)
(6)
The splitting convention in the previous example was dened by
an inner product on the graded Lie algebra L. We denote such a
graded Lie algebra by fL; [ ; ]; h ; ig: We shall always assume that
the splitting convention specied by fL; [ ; ]; h ; ig is given by
orthogonal complements with respect to h ; i: When this is the case
there is a simple characterization of those elements in normal form
(which we have not seen elsewhere).
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
33
Proposition 4.35. Suppose we are given a graded Lie algebra with
graded inner product, fL; [ ; ]; h ; ig: Then an element ` = ` +
` + 2 L is in normal form (to order s 1) if and only if the
following property holds for all 1 p ( s): if g 2 G and [g; `] =
mp + mp + then mp is perpendicular to `p . Furthermore each
orbit of the action of G = F L contains a unique representative in
normal form.
0
1
+1
1
Proof :
) When [g; `] = mp + mp
+ we have g 2 Cp (`), hence mp 2
Vp (`) = Yp(`)?. But ` in normal form means `p 2 Yp(`), and the
asserted condition follows.
( For any mp 2 Vp (`) Lp there is (by denition) an element
g 2 G such that [g; `] = mp + . The given hypothesis therefore
implies `p Vp (`)? = Yp(`), and we conclude that ` is in normal
form (to order s).
Existence and uniqueness was established in Theorem 4.20.
+1
1
1
1
1
q:e:d:
34
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
5.
Initial Linearity
Throughout the section M and N are Z-graded R-modules with
associated ltrations fF pM g and fF p N g, and cosets of submodules
are indicated with brackets. We assume that F M is a group w.r.t.
a binary operation possibly distinct from + , and we dene G :=
(F M; ). We assume in addition that F pG := F pM for p 1 denes
a ltration of G by subgroups.
1
1
For our purposes the appropriate general setting for the normal form
algorithm is an action of a ltered group G on a ltered vector space
having the property that the representation of each element g 2 G is
\linear modulo higher ltrations". Here we make this idea precise.
Denition 5.1. A (set-theoretic) mapping f : M
linear if it preserves the ltrations, i.e.,
(5.2)
f (F pM ) F pN
for all
!N
is initially
p 2 Z;
and has the form
f = fL + fH ;
(5.3)
were fL ; fH : M ! N also preserve the ltrations, fL is R-linear, and
for each (m; p) 2 M Z the following condition holds:
(5.4) 0 = [fL (m)] 2 N=F p N
)
0 = [fH (m)] 2 N=F p N :
+1
The subscripts L and H in (5.3) represent \linear" and \higher
order" respectively. Note that when f is R-linear it is initially linear :
take fL := f and fH := 0.
There is no requirement that the decomposition (5.3) be unique,
nor that fH be non-linear. However, when discussing initially linear
mappings a xed decomposition is always assumed.
For the remainder of the section we let ' : (g; n) 2 GN 7! g n 2 N
denote a ltration-preserving left action of G on N , i.e., an action such
that
(5.5)
F iG F j N
F i jN
+
for all (i; j ) 2 Z
+
Z:
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
35
Denition 5.6. We say that the action ' : G N ! N is initially linear if for each ` 2 F N the mapping f ` : G ! N dened by
f ` : g 7! g ` ` is initially linear.
0
Examples 5.7. Let L be a Z-graded R-Lie algebra with Lj the
trivial module for j < 0.
(a) For any ` 2 F L the mapping ad(`) : L ! L is linear, hence
initially linear with ad(`)L = ad(`).
(b) Assuming R is a eld of characteristic zero let G := F L with
0
1
the Campbell-Hausdor product. Then the expad action of G on
L is initially linear. Indeed, from (4.5) it follows that f ` : g 7!
expad(g )` ` is initially linear with fL` = ad(`) : g 7! [g; `].
(c) Assume R is a eld of characteristic zero and let n be a positive
integer. Then the collection gl(n; R) of n n matrices with
entries in R is a R-Lie algebra w.r.t.the usual matrix commutator
and becomes a Z-graded Lie algebra by taking gl(n; R)i to be
those matrices (mpq ) 2 gl(n; R) satisfying mpq = 0 if q p 6= i,
with the understanding that this refers to the zero matrix when
jij n.
Take L := T (n), where T (n) gl(n; R) is as in Example
2.5(a). Then G := F L acts on gl(n; R) via the expad mapping,
and by adapting the argument leading to (i) of Example 4.7 one
sees that expad(M )B = eM Be M . Since F L is invariant under
this action there is an induced action of G on the quotient (vector)
space N := gl(n; R)=F L. This quotient is not a R-Lie algebra,
since F L is not a Lie ideal of gl(n; R), but it does inherit a
Z-grading via Ni := (gl(n; R)i n ) for n i 2n
1.
The shift in indexing is to satisfy the ltration conditions in the
denition of an initially linear group action. One checks easily
that the action of G on N is ltration-preserving.
The quotient space N can obviously be identied with the
Lie subalgebra TL (n) gl(n; R) consisting of lower triangular
matrices (with non-zero diagonal elements allowed), and the induced action of G can then be described as follows: for M 2 G
and N 2 N ' TL (n) we have M N := (eM Ne M ), where
: gl(n; R) ! TL (n) replaces all entries above the diagonal of a
1
1
1
1
(2
1)
36
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
given matrix with zeros. Equivalently:
M N := N + [M; N ] + [M; [M; N ]] + :
1
2!
It is a simple matter to check that action (M; N )
M N 2 N is initially linear if we take
2 G N 7!
2 G 7! ( Æ ad(N ))M 2 N :
(d) Take R = C ; dene Lp = gl(n; C ) z p for all p 2 Z
Q
set L := [p2Z qp Lq . Dene the bracket of Az p 2 Lp
Bz q 2 Lq by
fLN : M
and
and
[Az p ; Bz q ] = [A; B ]z p q ;
+
where [A; B ] := AB BA is the usual matrix commutator, and
L is thereby given the structure of a graded Lie algebra. We think
of the elements as formal Laurent series
A(z ) = A p z p + + A z + A + A z + 1
1
0
1
in (the complex variable) z with coeÆcients in gl(n; C ).
Set G := F L, with the Campbell-Hausdor group structure.
Dene an action of G on L by g ` = expad( g )` + dzd g . (The
derivative represents formal term-by-term dierentiation of a series). This action is initially linear with fL` : m ! [`; m] + dzd m
provided one appropriately modies the denition of \initially linear action" to take into account the negatively graded terms. We
will not peruse this here.
This example arises when normalizing a rst order system y 0 =
A(z )y of meromorphic ordinary dierential equations on C at a
singularity, w.l.o.g. 0. Specically, the substitution y = P w =
(P (z )) w converts this equation to w 0 = (P A(z )P +P 0 P )w,
and
one
checks
that
(P; A(z ))
7!
P A(z )P +P 0 P denes a left action of Gl(n; C ((z ))) on gl(n; C ((z ))),
where C ((z )) is the quotient eld of the formal power series ring
C [[z ]]. This is the action by gauge transformations. To achieve
our context take P = eg .
1
1
1
1
1
1
1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
37
(e) An R-Lie algebra, M = (M; [ ; ]) is cyclically graded (of order
L
t) if M is the internal direct sum tj Mj of R-subspaces
satisfying
1
=0
[Mp; Mq ] = Mp q ;
p; q 2 Z=tZ :
+
To see an example let n > 0 be an odd integer and let
the collection of 2n 2n real matrices of the form
M=
A
T
S
A
M
be
;
were A = (aij ); S = (sij ); T = (tij ) 2 gl(n; K ) and S and T
are symmetric. This is an R -Lie algebra with the usual matrix
commutator as bracket, and becomes cyclically graded of order
4n 3 if we dene a grading as follows:
for 0 p n 1 and 3n 1 p 4n 3 we let Mp
consist of those M with the only non-zero entries, if any,
being elements aij of A satisfying p = j i.
for n p 3n 2 we let Mp consist of those M with
the only non-zero entries, if any, being elements sij of S
satisfying p = 3n (i + j ) and/or elements tij of T satisfying
p = n 2 + (i + j ).
The cyclicity property is easily veried.
The diÆculty with normalization in this context is \wrap around",
i.e., attempts to normalize a term `s 2 Ms in the inductive spirit
of the normal form algorithm can aect \lower order terms" (e.g.,
terms in Ms ) which have already been normalized.
We can circumvent the wrap-around problem as follows, assuming V is a (Z=tZ)-cyclically graded vector space (e.g. V := M
as above). We lift V to a Z-graded vector space Ve by dening
Vep := Vp z p ; where the subscript p on Vp is taken mod t, but
that on Vep, and the exponent in z p , is in Z. We think of the
elements in Ve as formal Laurent series
1
A(z ) = A p z p + + A z + A + A z + 1
1
where Ap 2 Vp .
0
1
38
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
We can now endow V with the structure of a graded R-Lie
algebra by dening the bracket of Az p 2 Vep and Bz q 2 Veq by
[Az p ; Bz q ] := [A; B ]z p q :
+
Example (d) above can be viewed as a special case of this construction: regard gl(n; C ) as a cyclically graded Lie algebra of
order 1 .
We will study cyclically graded Lie algebras in subsequent paper.
For later reference we record a few elementary properties of initially
linear mappings.
Proposition 5.8. For any initially linear mapping f : M ! N and
any m; m
^ 2 M the following properties hold :
(a) fL (m) = 0 ) fH (m) = f (m) = 0 ;
(b) m 2 F pM ) [f (m)] = [fL (m)] 2 N=F p N ;
(c) the condition 0 = [fL (m)] 2 N=F pN implies 0 = [fH (m)] 2
N=F q N for all q p + 1 ;
(d) the condition 0 = [fL (m)] 2 N=F p N implies [f (m)] = [fL (m)] 2
N=F p N ; and
(e) Assume p is the smallest integer such that 0 6= [fL (m)] 2 N=F p N
and/or 0 6= [fL (m
^ )] 2 N=F p N . Then [f (m+m
^ )] = [fL (m+m
^ )] =
p
[fL (m)] + [fL (m
^ )] 2 N=F N .
+1
+1
+1
Assertion (e) explains the \initial linear" terminology: taking m
^ =
0 we see that as p increases the element f (m) 2 N , if non-zero, is
\rst detected" within the factor modules N=F pN as a value of a
linear mapping.
Proof :
(a) Immediate from (5.4).
(b) Immediate from the denition.
(c) Since the inclusions F pN F q N for p q induce epimorphisms
N=F pN ! N=F q N this is immediate from (5.4).
(d) By (c) and f = fL + fH .
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
39
(e) Replace m by m + m
^ in (5.4) and use the linearity of fL .
q:e:d:
The normal form denition given in x4, and the normal form algorithm seen in Proposition 4.19, generalize easily to the context of the
initially linear group action ' : G N ! N under consideration in this
section. Specically, given s 2 N and ` 2 F N dene vector spaces
Cs (`) and Vs (`) analogous to (4.12) and (4.13) as follows :
0
1
1
4
+1
(5.9)
Cs (`) := fg 2 Gjf ` (g ) 2 F s N g = fg 2 GjfL` (g ) 2 F s N g
1
+1
+1
and
Vs (`) = s (f ` (Cs (`))) = s (fL` (Cs (`))):
(5.10)
1
1
1
+1
+1
+1
Notice that Cs (`) = Cs (Js (`)) : Indeed with ` = Js (`) + b̀ we have
g 2 Cs (`) ,
g`=`
mod F s (N )
, g (Js(`) + b̀) = Js(`) + g b̀ mod F s (N )
,
g (Js (`)) = Js (`)
mod F s (N )
,
g 2 Cs (Js(`))
As a consequence we see that Vs (`) = Vs (Js (`)) .
Now assume a splitting convention, i.e., that for each s 1 a complement Ys(`) Ns of Vs (`) has been chosen which depends only on
Js (`), hence that
1
1
1
+1
+1
+1
1
1
1
+1
+1
1
1
(5.11)
Ns = Ys(`) Vs (`);
s 1:
1
In particular,
(5.12)
,
Vs (`) = Ns
1
Ys(`) = 0:
To involve all non-negative indices in the denition of Vs (`) dene
1
Y (`) := N :
(5.13)
0
0
Denition 5.14. An element ` = ` + ` + 2 F N is in normal
form to order s 0 (w.r.t. the assumed splitting convention) if `j 2
Yj (`) for j = 0; : : : ; s, and is in normal form if it is in normal form
to order s for all s 0.
0
Recall that for each
g `
`.
4
`
2C
the mapping
0
1
f
`
:
G!
N
is dened by
f
`
:
g
!
40
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Proposition 5.15. Suppose ` = ` + + `s + is in normal form
to order s 0. Write `s = `Ys + `Vs in accordance with the
decomposition (5.11) (with s replaced by s + 1). Choose g 2 Cs (`)
such that s f ` g = `Vs . Then g ` is in normal form to order
s + 1 and Js (g `) = Js (`).
0
+1
+1
+1
1
+1
+1
The astute reader may have noticed that the negative sign in the
equality s f ` g = `Vs of the preceding statement does not appear explicitly in the normal form algorithm described in x4. It does,
however, appear surreptitiously: Cs (`) is dened in terms of ad(`)(g ) =
[`; g ], and expad(g )(`) has initially linear term [g; `] = ad(`)(g ):
+1
+1
1
Proof : We have
g` = `+g` `
= ` + + `s + `s + f `(g ) + fterms in F s N g
= ` + + `s + `Ys + `Vs
`Vs + fterms in F s N g
= ` + + `s + `Ys + fterms in F s N g;
0
+1
0
+1
0
+1
+2
+2
+1
+1
+2
which by Ys (`) = Ys (g `) is in normal form to order s + 1. q.e.d.
+1
+1
Proposition 5.16. Suppose ` = ` + ` + 2 F N and `;^ ` 2 F N
are elements in the G -orbit of ` in normal form to order s 0. Then
Js (`^) = Js(`).
0
0
1
0
In other words: the normal form of ` is unique to all orders.
Proof : It is enough to deal with the case ` = `, and this we do by
means of induction on s 0. By assumption there is a g 2 G = F M
such that
1
g ` = `:^
(i)
To verify the case s = 0 write
`^ = g ` = ` + (g ` `) = ` + f `(g ):
Since f ` preserves ltrations and g 2 F G we see that `^ = ` +
fterms in F N g, and this case is established.
1
1
0
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
41
Now assume s 0, that uniqueness holds for s, and write
` = Js (`) + `s + fterms in F s N g;
`^ = Js (`^) + `^s + fterms in F s N g
= Js (`) + `^s + fterms in F s N g :
From (i) and the equality of the s-jets we have g 2 Cs (`), and by
the initial linearity assumption we have
`^ = Js (`) + (`s + fL` (g )s ) + fterms in F s N g;
+1
+1
+1
+1
+1
+1
1
+2
+1
+1
hence `s = `^s + fL` (g )s , i.e., `s
`^s = fL` (g )s . However, by
denition we have fL` (g )s 2 Vs (`), whereas `s
`^s 2 Ys (`)
by the normal form assumption, and `s = `^s follows.
q.e.d.
+1
+1
+1
+1
+1
+1
+1
1
+1
+1
+1
+1
+1
+1
42
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
The Spectral Sequence of an Orbit of an Initially
Linear Group Action
Throughout the section R is a eld and G and L are respectively
Z and Z-graded vector spaces over R. We suppose G is also a
group, with binary operation , having the property that the ltration
fF Gg 2Z of G as a vector space also provides a ltration of G as
a group. Finally, we assume ' : (g; `) 7! g ` is a left action of G on
L which is initially linear in the sense of Denition 5.6, i.e. for each
` 2 L the mapping
(6.1)
f :G!L
6.
+
p
p
+
`
dened by
(6.2)
f ` : g 2 G 7! g ` ` 2 L
is initially linear.
As remarked in the introduction any orbit O of ' can be viewed
as a category OC : objects are the points ` 2 O ; morphisms between
objects `; `^ are elements g 2 G such that g ` = `^; compositions are
dened by multiplication within G .
With a minor additional hypothesis we can dene a covariant functor
from each orbit OC to the category of spectral sequences. The hypothesis is needed to further relate the group and vector space structures of
G . For each g 2 G let cg : a 2 G ! g a g 2 G denote conjugation
by g 2 G . We assume cg is ltration preserving. This is easily seen to
be the case if G is given by the Campbell-Hausdor formula.
1
Assumption 6.3. cg (a b) = cg (a) + cg (b) 2 F p G =F p
Z and all a; b 2 F p G .
+1
G
for all p 2
+
When the group structure is induced by the Campbell-Hausdor
formula, as in all the examples of the previous sections, the assumption
is an easy consequence of the identity x = x. Indeed, in this
context each cg is induces the identity mapping on F p G =F p G :
Our functor will be a composition. To dene the initial factor associate to each ` 2 OC the sequence
1
+1
(6.4)
0!G
!L!0
f`
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
43
and to each morphism g 2 OC the commutative diagram
L ! 0
#g
(6.5)
f 0
! L ! 0
For the second factor recall from x3 that there is a spectral sequence
corresponding to each linear mapping h : G ! L, and we can therefore
associate with each object (6.4) the spectral sequence fErp;q (`)g of the
0
! G
# cg
! G
!
f`
g `
linear mapping
fL` : G ! L:
(6.6)
Now observe, from Assumption 6.3, that the mappings induced by
the morphisms (6.5) are linear in the quotients dening these spectral
sequences, and as a result we obtain a functor from the orbit category
OC to the category of spectral sequences.
It is worth noting that the spectral sequences can be dened directly
from the objects ` 2 OC , whereas the morphisms require the introduction of the intermediate category. In classical normal form calculations
this corresponds to working with ad(`) rather than expad(`) when
computing with the normal form algorithm.
With (4.27) as the motivating example we generalize denitions
(5.9) and (5.10).
Denitions 6.7. For p 1 and r 0 dene
(a) Crp (`) = fg 2 F p GjfL` (g ) 2 F p
+r
Lg
(b) Vrp (`) = p r (fL` (Crp (`))) Lp r :
+
and
+
We again have inclusions as seen in (4.26), i.e.,
(6.8)
Cp
1
+r
1
(`) C p
+r
2
2
(`) Crp(`) Cp
1
+r
1
(`) F
1
G;
and these in turn induce inclusions
(6.9)
Vp
+r
1
1
(`) V p
2
+r
2
(`) Vp
1
+r
1
(`) Lp r :
+
We are using the fact that G is an Z graded group to conclude that
the above sequences of inclusions are nite. The terms appearing in
the spectral sequence fErp;q (`)g are easily seen to be related to the
+
44
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
R-modules appearing in (6.7) as follows:
(6.10)
8
(a) Zrp; p(`) = Crp (`) F p G ;
>
>
>
< (b) E p; p (`) (C p (`)) L ; where : L ! L denotes
p
p
p
p
r
r
>
the projection, and
>
>
:
p; p
(c) Er
(`) Lp=Vrp r (`):
We claim that Crp (`) is a subgroup of G . Indeed, for g 2 G we
have g 2 Crp(`) if and only if g 2 Lp and g ` = ` modulo F p r G . If
a; b 2 Crp (`) then (a b) ` = a (b `) = a ` = ` modulo F p r G , and
the subgroup assertion follows.
+1
+
+
Theorem 6.11. Assuming the standing hypotheses of the section the
following entities are invariants of any xed G -orbit :
(a) the spectral sequences fErs;t(`)gr ;
(b) the vector spaces p (Cpq (`)) ;
(c) the factor spaces Lp r =Vrp (`) ;
(d) the vector spaces Vrp(`) ;
(e) the subgroups Cpq (`) .
Moreover, each spectral sequence fErs;t(`)g is strongly convergent and
for each p 1 we have
0
+
(ii)
p; p
E1
= p (f g 2 F pG j fL` (g ) = 0 g)
and
(ii)
p;
E1
p+1
= Lp=Vp (`) :
1
1
Finally, when the conjugation mappings cg induce the identity mappings on each F p G =F p G the isomorphisms associated with each of
the invariants in (a)-(e) are given by the identity mapping.
+1
In the statement Assumption 6.3 is included among the standing
hypotheses. Also recall, from x2, that the vector spaces Gp and Lp
are assumed nite-dimensional.
Proof :
(a), (b) and (c) : Diagram (6.5) induces an isomorphism of spectral
sequences with inverse induced by the action of g . The isomorphisms
now follow from (6.10).
1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
45
(d) : The isomorphism in part (c) is induced by the action of G . For
g 2 G the diagram
0
!
Vr
0
!
p
Vr g
p
(
!
(`)
)
`
!
F
F
p+r
p+r
L ?
?
=F
p+r +1
y
L
=F
LL
!
LL
!
p+r
p+r;
(p+r )+1
?
?
y
Er
g
p+r +1
L
(`)
()
p
p+r =Vr `
p+r
(p+r )+1
( g `)
L
(
p
p+r =Vr g
(e) : It suÆces to show that cg : Crp (`) ! Crp(g `) is dened. However,
for a 2 Crp(`) we have cg (a) (g `) = g a g (g `) = g (a `) = g `
modulo F p r L, implying cg (a) 2 Crp(g `).
1
+
For the nal convergence statement use (c) and (6.9) and for (i) the
nite dimensionality of G .
q:e:d:
The spectral sequence chart may help clarify the convergence. The
dierentials, dr originating in position (p; p) must eventually be zero
because E p; p is nitely generated. For r large Erp; p is not in the
image of a dierential because the ltration of G is bounded below
by 1 . Notice that the nite-generation hypothesis on the Gp and Lp
(originally stated in x2) is not needed to deduce strong convergence in
positions (p; p + 1).
To detail the connection between the spectral sequence computations and the algorithm in x4 express diagram (3.16) in terms of the
equivalences of (6.10):
+1
0
!
`
fL
Crp (`)
p;r
#
Fp
+r
#
L
p+r;r
! Lp r =Vrp(`)
Erp; p(`)
dr
+
Note that when the action is expad, ` = ` s is in normal form to
order s 1 and r = s p + 1 this becomes
`( ) j
( ( ))
p
s
Cs p (` )
! +1
Fs L
(6.13)
+1 #
# +1 +1
( )
ad(
( )
s
)
p
s
C
p
` s
+1
+1
p;s
p
s
Esp; pp (` s )
( )
+1
0
!
0
g
p+r;
Er
can therefore be completed to a commutative diagram of short exact
sequences, and the resulting R-linear map Vrp (`) ! Vrp (g `) must be
an isomorphism by the 5-lemma.
(6.12)
!
ds
!
p+1
;s
p
Ls =Vsp p (` s )
+1
( )
+1
`
)
46
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
The connection is now transparent: the method for constructing normal
forms introduced in x4 emphasizes the top line of this last commutative
diagram; the spectral sequence approach emphasizes the bottom line.
From Theorem 6.11 we see that this bottom line can always be
computed by replacing ` s with the original element ` 2 L to be
normalized. In particular, one does not have to successively introduce
the partially normalized elements ` s to do the calculations. This
justies dropping ` from the notation, and we do so when confusion
cannot otherwise result, i.e., we simply write that bottom line as
( )
( )
Esp; pp
ds
+1
! Ls =Vsp p :
p+1
+1
+1
To further ease notation we generally express
Proposition 6.14. For any v
equivalent:
2 Fs L
+1
Lp=Vp
1
1
as L=V , etc.
the following statements are
(a) s (v ) 2 Vsp p (`); and
+1
+1
(b) [v ] := s
+1;s
p+1
(v ) is killed by the dierential ds
p+1
.
Distinctions between the two assertions, as well as notational distinctions between elements of L=V and their representatives in V ,
are often blurred. For example, either of (a) and (b) might be indicated by any one of the following statements: p (v ) is killed by the
dierential; p (v ) is killed by ad(`); (the class) v is killed by the
dierential; and (the class) v is killed by ad(`).
Proof : For the expad-action this is clear from the commutativity
of (6.13); the argument for the general case is completely analogous.
q:e:d:
The concept of an initial term (see (4.28)) generalizes in the obvious
way to the context of an initially linear group action. Specically, the
initial terms of the subgroup Csp p (`) G are dened by
+1
(6.15)
Isp
p+1
(`) := p (Csp
p+1
(`)) Lp ;
and an element mp 2 Isp p (`) is said to complete in Csp p (`). Comparing (6.15) with (6.10c) and assuming the expad-action we see that
+1
+1
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
47
diagram (6.13) can now be written
(6.16)
p
j
C
p
s p+1
p
s
C
p+1
(`(s) )
Isp
p+1
ad(`
(` )
(s)
j
(s) )
#
ds
(` s )
( )
!
p
s
C
p+1
!
(`(s) )
+1
#
p+1
Proposition 6.17. For any element mp
ments are equivalent:
(a) mp completes in Csp p (`);
(b) p (mp ) 2 Isp p (`); and
Fs
s+1;s
p+1
Ls =Vsp p (` s )
+1
2 Lp
L
( )
+1
the following state-
+1
+1
(c) mp survives to Esp; pp (`).
+1
Proof : In the case of the expad-action use the commutativity of (6.16)
in combination with Proposition 4.32(d); the proof for the general case
is completely analogous.
q.e.d.
We have remarked in x4 that in normal form calculations the spaces
Vp (`) can be diÆcult to compute. We now see this as an artifact
of the method used. Indeed, it is evident from (6.12) that from the
spectral sequence viewpoint one should simply compute the E1 term
p; p
E1
= Lp=Vp and then realize Vp as the kernel of the canonical
linear mapping p : Lp ! Lp =Vp . This factor space philosophy also
carries over to splitting conventions: a complement Y Lp of Vp
must be the image of a section s : Lp=Vp ! Lp of p , and from this
one sees that to determine Y from Lp =Vp it is only necessary to
specify that section. Finally, the conversion of a given ` 2 L to normal
form can now be regarded as killing successive terms of ` (s Æ )`, and
this can be accomplished via (6.12) and Theorem 6.11 in terms of the
dierentials computed directly from the initially given `. In particular,
the information buried in the dierentials is more than suÆcient to
calculate the normal form.
In the following examples the actions are derived from the expadaction, and as a consequence the induced mappings of spectral sequences are the identity (see the remark immediately following the
statement of Assumption 6.3). It follows that the calculations depend
only on the orbits of G .
1
1
+1
1
1
1
1
1
1
1
1
1
1
1
1
48
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Example 6.18. We rework Example 4.34 using the spectral sequence
approach to normalization. The matrix to be normalized was
0
0
B0
B
B0
B
0
` := B
B
B0
B
B0
@0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
4
0
0
1
0
0
0
0
0
0
3
0
1
0
0
0
6
0
8
0
0
0
0
0
1
7
12C
C
0C
0C
C
;
0C
C
2C
C
0A
0
the relevant spectral sequence, i.e., that induced by the linear mapping
ad(`) : G ! L, was already computed in x3. (Keep in mind that
f ` : g 7! expad(`)(g ) ` is only needed to understand morphisms of
spectral sequences; the linear term fL` : g 7! ad(`)g alone suÆces to
compute the actual spectral sequence.)
For purposes of dening the normal form we use the same splitting
convention as in Example 4.34, i.e., we take orthogonal complements
w.r.t. the inner product hA; B i := tr(A B ).
From the work in Example 3.18 we know that the only non-trivial
spaces of the E1-term L=V in ltrations greater than 1 are L =V
and L =V , hence Lj = Vj for j = 3; 4; 5 and 7. Without any
additional work we can conclude that the normal form `N = `N + `N +
+ `N of ` must have `Nj = 0 for these particular values of j . We
also know from Example 3.18 that each of L =V and L =V has
a single generator, i.e., the images ! and ! under of e and
e respectively. This information is conveniently summarized by the
diagram
1
2
6
1
5
1
1
1
0
1
7
1
2
2
1
6
1
5
6
26
61
L
L
#"s
R f! g
2
(i)
2
L
#
L
3
L
#
4
0
0
L
L
#
L
L
L
#"s
R f! g
5
6
0
L
#
7
0
6
wherein the bottom row represents L=V and s is the section of
: L ! L=V uniquely determined by the condition s(L=V ) = Y .
The rst class that needs to be killed is ` s` = 3e , and from
the calculation of the E -terms in Example 3.18 we see that e =
3(e + e ) does the job (as does 3e , which is the matrix M used
3
3
33
2
23
22
1
24
2
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
49
in the Order 3 calculation of Example 4.34). We have
0
`
(3)
0
B0
B
B0
B
0
:= expad(e )(`) = B
B
B0
B
B0
@0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
4
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
6
0
8
0
0
0
0
0
1
7
12C
C
0C
0C
C
;
0C
C
2C
C
0A
0
which is in normal form to order 3, but this is not immediately relevant:
we continue working with the original ` and use the dierentials in the
spectral sequence to produce matrices e = 4e + 8e , e = 12e
and e =
e which kill the remaining (` (s Æ )`)-terms. The
normal form
0
1
0 0 0 0 0 0 6 0
B0 0 1 0 0 0 0 0C
B
C
B0 0 0 1 0 0 0 0C
B
C
B0 0 0 0 1 0 0 0C
B
C
B0 0 0 0 0 1 0 0C
B
C
B0 0 0 0 0 0 0 2C
@0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
of ` is then obtained by conjugating ` by ee4 e3 e2 e1 = ee4 ee3 ee2 ee1 :
2
31
34
3
53
7
4
2
51
Note that the normal form ` to order 3 obtained above is not
(quite) the same as the analogous ` obtained in Example 4.34, although both have the same 3-jet, thereby illustrating uniqueness up to
order three. The full normal forms do coincide.
(3)
(3)
Example 6.19. We next illustrate the spectral sequence approach to
normal forms by applying the methods to an example of the type described in Example 5.7(c), here taking n = 5. Recall gl(5; R) is
a Z -graded Lie algebra with gl(5; R)i consisting of those matrices
epq 2 gl(5; R) satisfying epq = 0 if q p 6= i . In particular gl(5; R)i = 0
if i does not satisfy 4 i 4: Note that G := F L where L = T (5),
the upper triangular matrices. N := gl(5; R)=F L which may be identied with TL (5), the lower triangular matrices (with non-zero diagonal
allowed). N is graded by Ni := (gl(5; R)i ); 5 i 9:
The matrix we will analyze is
1
1
9
50
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
0
1
1 0 0 0 0
B2 1 0 0 0C
B
C
C:
1
2
1
0
0
`=B
B
C
@0 4 2 1 0A
0 6 3 11 1
This is of some interest because it is nongeneric
0 , i.e., the
1 lower-left
1 2 1
0 4
subdeterminants det 0 ; det 0 6 and det @0 4 2A all vanish
0 6 3
[GR]. Our notation follows the previous example with modications to
adjust for the ltration shift in N .
The basis we use for G s is given by fesk g; 1 s 5; 1 k 5 s ,
where esk is the 5 5 matrix with ak;k s = 1 and all other entries 0.
The basis for N s is given by fesk g; 5 s 9; 1 k s 4 , where
esk the 5 5 matrix with a s k;k = 1 and all other entries 0.
The example was also chosen to illustrate some of the subtleties that
arise when passing from the spectral sequence to the normal form. The
E term is displayed by the following chart.
+
9
+
2
0
0
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
@@ @@
@ @
@@4 R @@
@@ 3 R@@
@@ @@
@2R @
@@ @@
@@1R @@1 R
@@ @@ 2 R
@ @
@@ @@ 3R
@ @
@@ @@ 4 R
@ @
@@ @@ 5 R
@ @
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
51
We compute the dierentials as we did in the previous example. The
rst non trivial dierential is a d . The calculation of E s;t is similar to
the previous example and is left to the reader. The only non trivial d 's
are: d (e ) = 6e ; d (e ) = 6e ; d (e ) = 6e ; d (e ) =
6e and d (e ) = 6e + 6e . Hence
6
7
6
6
12
82
73
6
E;
E;
E;
E;
E;
E;
1
84
6
23
95
13
= R f[e ]g
= R f[e ]; [e ]g
= R f[e ]; [e ]; [e ]; [e ]g
6
71
7
7
9
22
31
7
8
6
21
3
7
7
72
92
11
2
7
3
32
14
= R f[e ]; [e ]g
= R f[e ]g
= R f[e ]g
1
7
2
6
8
7
The class [e ]
are d 's:
95
81
83
91
92
2E;
9
93
94
was set equal to [e ] by a dierential. There
8
92
7
7
[e ] !
7 [e ]; [e ] !
7 [ 4e
[e ] !
7 [ e +e ]
11
82
21
13
91
82
+ 3e ]
84
93
The rst two dierentials dened on ltration 1 are zero in E and as a
result we see that [e ] and [e ] survive to E . In E the element [e ]
is represented by e + e : The precise identication of the representative of [e ] 2 E is necessary for computing d ([e ]). This is related
to Proposition 6.17, but is perhaps most easily explained in terms of
the discussion of completions beginning just before Proposition 4.32.
Specically, in the language of spectral sequences the calculation of
ad(` )(F L) + F L beginning immediately before (v) in Example
4.34 amounts to calculating d , and the discussion following the computation of ad(` )(m ) is related to computing d . The fact that one
may choose a matrix in the image of ad(` ) which is also in the image
of ad(` )(F L) allows us to complete a choice of m to a matrix in
C (` ), which from the spectral sequence perspective shows that m
survives to E . Hopefully this attempt to relate the calculations above
to those in x4 has enlightened rather than confused the reader. A similar argument shows that [e ] survives to E and is represented by
e
e
e . d may now be determined:
7
11
13
8
8
11
1
11
11
(5)
23
6
8
5
8
11
6
1
(5)
4
2
(5)
(5)
4
5
4
(5)
4
2
3
13
13
3
8
1
2
23
2
22
[e ] !
7 [ 2e
[e ] 7! [2e
11
13
92
8
91
+ 2e
2e ]
92
95
1
2
e + e =
1
93
2
95
5
2
e + e;]
5
91
2
9 2
52
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
[e ] survives to E and [e ] = [e ] 2 E .
p; p
The resulting E1
is summarized by a chart analogous to (i) of
the previous example:
13
9
91
92
9
+1
(6.20)
L
L
k
R f! g
L
L
k
5
L
R
5
f! ; ! g
(1)
(2)
6
6
L
L
#"s
R f! g
6
7
R
7
L
#"s
8
f! ; ! g
(1)
(2)
8
8
L
#"s
9
R
f! ; ! g
(1)
(2)
9
9
where
s(! ) = e
s(! ) = e ; s(! ) = e
s(! ) = (e ; + e + e ; + e ); s(! ) = e ;
(e ) = (e ) = 0
(e ;) = (e ) = 0
(e ) = (e ) = (e ) = (e ) = !
7
71
(1)
(2)
81
8
1
1
9
4
72
8
83
8
9 1
92
9 3
2
95
94
9
73
84
(1)
91
92
93
95
9
(The splitting is dened as in the previous example.)
We now use the dierentials in the spectral sequence to convert ` to
normal form, rst noting that ` is already in normal form to order 6.
In degree 7 we have to kill
0
0
B0
B
B0
B
@0
0
0
0
0
4
0
0
0
0
0
3
0
0
0
0
0
1
0
0C
C
0C
C
0A
0
(this follows from (6.20)), and by computing the dierential d one
sees that the matrix
6
0
0
B0
B
m =B
B0
@0
0
1
0 0
0 1=2
0 0
0 0
0 0
0
0
0
0
0
1
0
0 C
C
0 C
C
2=3A
0
will do the job. So the 7th normal form (= m `) is
0
`
(7)
1
B5=2
B
=B
B 1
@ 0
0
1
0
2
2
0
6
0
0
0
0
0
1
0
0
0
0 C
C
0
0 C
C
19=3 0 A
11 25=3
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
53
We have, leaving the details to the reader:
1
0
1 0 0
0
0
B5=2 2 0
0
0 C
C
B
0
0 C
` =B
C
B 1 0 0
@ 0 0 0
19=3 0 A
0 6 0
0
25=3
(8)
where `
(8)
=m ` ,
(7)
2
0
0
B0
B
m =B
B0
@0
0
0
0
0
0
0
2
1
0 0
0 11=6
0 0
0 0
0 0
0
0 C
C
1=3C
C
0 A
0
The nal step is to nd ` . The term in degree 9 of ` has the form
[ (e + e + e + e )
e ]+( e
e
e + e ),
where the term in the parenthesis can be killed by a dierential. (The
terms are enclosed in square brackets and parenthesis to distinguish
the components in the splitting. Specically we have written ` =
[s (`)] + (` s (`)) :) From this point the unique normal form
1
0
0 0 0 0
B5=2
0 0 0C
C
B
B
0 0C
` =B 1 0
C
@ 0
0 0
0A
0 6 0 0
is achieved with very little eort; nding the matrix that transforms
` into ` requires a bit more work.
First note that
(9)
17
6
(8)
19
91
92
93
95
3
11
94
5
91
6
6
17
92
6
33
93
6
95
17
6
17
6
(9)
17
6
19
3
17
6
(8)
11
e
6 91
(9)
5
e
6 92
17
33
e93 +
e
=
6
6 95
28
(e
6 91
e92
)
17
(e
6 93
e91
)+
11
(e
3 95
e92
);
and that the terms in the parenthesis in the right side of the equality
are hit by dierentials, e.g.,
(e + e ) 7!
(e
e ): In this
way we determine that the matrix
1
0
0
0 0
C
B0 0
0 0
C
B
B
m = B0 0 0 0 C
C
@0 0
0 0 0A
0 0 0 0 0
28
15
1
11
6
28
17
15
6
28
23
6
11
12
14
3
satises m `
3
(8)
=` .
(9)
45
91
92
54
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
We now illustrate Theorem 6.11. If we compute the dierentials
in the spectral sequence Er; (` ) we nd d (e ) = 0, d (e ) = 0
and d (e ) =
e + e . In the quotients that dene Er; these
dierentials are identical to the corresponding dierentials in Er; (`).
(9)
5
8
11
2
7
13
8
13
5
91
2
92
We conclude with a trick which, in some cases, may be used to
compute a large part of the normal form without having to determine
the transforming matrices. From (6.20) we know that there must be
real numbers bij ; a and b such that
0
1
a 0 0 0 0
Bb21 a 0 0 0C
B
C
C
N =B
Bb31 0 a 0 0C
@ 0 0 b43 b 0A
0 6 0 0 a
is the normal form. Now compute the spectral sequence Er; (N ). The
invariance of this sequence, in particular that of the dierentials, completely determines the normal form to order 8. (Unfortunately, we
cannot determine the diagonal elements in this manner.)
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS
55
References
[A] V.I. Arnold, Spectral Sequences for the reduction of Functions to Normal
Form, In Problems in Mechanics and Mathematical Physics (Russian), Izdat.
\Nauka", Moscow 297, (1976), 7-20.
[B] A. Baider, Unique normal forms for vector elds and Hamiltonians, J. Di.
Eqns., 78 (1989) 33-52.
[BC] A. Baider, R.C. Churchill, The Campbell-Hausdor group and a polar decomposition of graded algebra automorphisms, Pacic J. Math, 131, (1988),
219-235.
[C] K.-T. Chen, Equivalence and decomposition of vector elds about an elementary critical point, Am. J. Math. 85 (1963), 693-772.
[CK] R.C. Churchill and M. Kummer, A Unied Approach to Linear and Nonlinear
Norms Forms for Hamiltonian Systems, J. Symbolic Computation, 27 (1999),
49-131.
[CKR] R.C. Churchill, M. Kummer and D.L. Rod, On averaging, reduction and
symmetry in Hamiltonian systems, J. Dierential Equations, 49 (1983), 359414.
[GR] M.I. Gekhtman and L. Rodman, Normal forms of generic triangular band
matrices and Jordan forms of nilpotent completions, Linear Algebra Appl.,
308, (2000), no. 1-3, 1-29.
[Gode] R. Godement, Topologie Algebrique et Theorie des Faisceaux, Hermann,
Paris, 1958.
[J] N. Jacobson, \Lie Algebras", Dover Publications, Inc. 1962.
[Mac] S. MacLane, Homology, Academic Press, New York, 1963.
[Mur1] J. Murdock, Hypernormal form theory: Foundations and Algorithms, J.
Di. Eqns. 205, (2004), 424-465.
[Mur2] J. Murdock, \Normal Forms and Unfoldings for Local Dynamical Systems",
Springer-Verlag, New York, 2003.
[Sa1 ] J. Sanders, Normal form theory and spectral sequences, J. Di. Eqns., 192,
(2003), 536-552.
[Sa2 ] J. Sanders, Normal form in ltered Lie algebra representations, To appear in
Acta Applicandae Mathematiae.
[Se] J.P. Serre, \Lie Algebras and Lie Groups", Benjamin, New York, 1965.
[Sp] E. Spanier, \ Algebraic Topology", Springer, New York, 1966.
E-mail address : [email protected]
E-mail address : [email protected]
Hunter College and Graduate Center, CUNY, New York, NY 10021