A SPECTRAL SEQUENCE APPROACH TO NORMAL
FORMS
MARTIN BENDERSKY AND RICHARD C. CHURCHILL
1. Introduction
Normal forms for vector fields and Hamiltonians at equilibria have
a long history, an extensive literature, and a continuing appeal for
researchers (e.g., see the references in [Mur1 ], [Sa1 ]). 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: find 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 re-
lated to spectral sequence calculations (see Arnol'd [A ] for the case of
singularities; Sanders and Murdock [Sa1 ], [Mur1 ] for the case of vector
fields). Specifically, in [Sa1 ] Sanders showed how one could interpret
the normal form algorithm in terms of a minor variation of the standard
spectral sequence of a filtered module with a compatible grading (also
see [Sa2 ]). 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 different approach to construct
spectral sequences indexed by the elements `. This approach allows
____________
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 spec-
tral 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 associ-
ated with each orbit O of a group action ' : G x X ! X : the objects
are the points of the orbit; a morphism between objects X1, X2 is
an element g 2 G such that g . X1 = X2; composition is defined by
multiplication. When the action is initially linear, as defined in x5,
and when one additional hypothesis is satisfied, we construct a functor
from OC to a category of short cochain-complexes, thence to the cat-
egory 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. Specifically, x3 is included for the benefit 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 gen-
eralizes normal form theory to the action of a group on a vector space.
This goes beyond Baider's context and encompasses other widely stud-
ied "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 particu-
lar, we are now able to treat the one normal form case for indecompos-
able 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 afield from the lecture delivered by the first author at the
conference celebrating Sam Gitler's 70th birthday, it seems a fitting
illustration of the rich diversity of Sam's interests.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 3
2. Preliminaries
Throughout the paper R denotes a commutative ring with multi-
plicative identity 1 6= 0 and all modules are assumed (left) R-modules
unless otherwise stated. A filtration {F pM}p2Z of a module M (by
R-submodules F pM) will always refer to a decreasing filtration, i.e.,
(2.1) q > p ) F qM F pM.
When the inclusion in (2.1) holds we refer to F qM as a higher filtration
than F pM.
We will always deal with modules M having the following struc-
ture: {Mp}p2Z is a family of free modules of finite dimension, F pM :=
Q p
q pMq for each p 2 Z, and M := [p2ZF M. The construction
guarantees that elements m 2 M can be regarded as formal infinite
sums
(2.2) m = mq + mq+1 + . . . with mp 2 Mp ,
which for q < 0 one could think of as a Laurent series. Note that
{F pM}p2Z defines a filtration 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 pMp and the (direct) product pMp.)
For any p 2 Z the p-jet Jp(m) of m = mq + mq+1 + . . .2 M is
defined by
ae
mq + . .+.mp if p q
(2.3) Jp(m) :=
0 otherwise.
When a graded module M is also Lie algebra with bracket [ , ]
satisfying
(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 iLi and product
Q
iLi 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 |i| n. Condition (2.4) is easily verified. As
an illustration of jets: the 2-jet of an element
0 1
* * * * *
B 0 * * * * C
B C
m = BB 0 0 * * * CC2 L
@ 0 0 0 * * A
0 0 0 0 *
is given by
0 1
* * * 0 0
B 0 * * * 0 C
B C
J2(m) = BB 0 0 * * * CC,
@ 0 0 0 * * A
0 0 0 0 *
wherein corresponding entries in m and J2(m) indicated by aster-
isks are identical.
(b) Let K = R or C and let Vect(n) denote the K-space of for-
P n @
mal vector fields X = j=1pj ____ in equilibrium at 0, i.e., the
@xj
formal power series coefficients pj 2 K[[x1, . .,.xn]] are without
constant terms. Vect(n) is given the structure of a K-Lie alge-
P n @
bra by defining the bracket of elements X = j=1pj ____ and
@xj
P @
Y = j qj ____ to be
@xj
_ !
X X @qj @pj @
[X, Y ] := (pi____- qi____) ____ .
j i @xi @xi @xj
It is given the structure of a graded Lie algebra by setting Vect i(n) :=
P n @pj
0 when i < 0 and letting Vect i(n) denote those X = j=1 pj____
@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 fields at equilibria is one of the standard
applications of normal form theory (see, e.g., [Mur1 ] and [Sa1 ]).
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 differential object consists of a module E together with an R-
linear mapping d : E ! E, known as the differential, satisfying d2 = 0.
Any cochain complex
q-1q ffiqq+1
(3.1) . .!.Eq-1 ffi-!E -! E ! . . .
L
can be considered a differential object: take E := qEq and define
P P
d : E ! E by qeq 7! q ffiqeq. Indeed, alternate notation for (3.1),
which we immediately adopt, is
(3.2) . . .! Eq-1 - d! Eq -d! Eq+1 ! . ...
Similarly, any chain complex may be considered a differential object.
L
Another important example is the direct sum E := (p,q)Ep,q of
R-modules indexed by Z x Z together with a differential d : E ! E
satisfying d|Ep,q : Ep,q ! Ep+r,q-r+1 for all p, q . In this instance
the differential object is called a bigraded module with differential of
bidegree (r, -r + 1) (e.g., see the spectral sequence charts in Example
3.18).
The derived module H(E) of a differential object (E, d) is defined
by
(3.3) H(E) := ker{d : E ! E}=dE ;
this module is also called the cohomology (resp. homology) of E, par-
ticularly in the case of a cochain (resp. chain) complex.
A spectral sequence is a sequence {(Er, dr)}1r=0of differential objects
such that Er+1 ' H(Er) for all r. In the latter definition no relation-
ship between the various differentials is assumed, although in prac-
tice they are often induced by the same mapping. We follow custom
and express the R-module isomorphisms Er+1 ' H(Er) as equalities.
Moreover, when confusion cannot otherwise result we write all dr and
all restrictions thereof as d.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 7
__ __
A map (or morphism) f : {(Er, dr)}1r=0! {(E r, dr)}1r=0 of spectral
__
sequences is a collection of R-linear mappings fr : Er ! E r com-
__
muting with the differentials, i.e., satisfying fr O dr = dr O fr for all
r 0 .
Suppose {(Er, dr)}r 0 is a spectral sequence and k 0 is an integer.
An element e 2 Ek survives to Ek+1 if e 2 ker dk, in which case e
determines a coset [e]k+1 2 Ek+1 = 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+1 of [e]k+r in
Ek+r+1 is seldom represented by e (as we will see in examples). All we
can say is that [e]k+r+1 is represented by an element with leading term
e in lowest filtration. An element e 2 Ek is killed if e 2 dEk. Notice
from d2k= 0 that such a class must survive to Ek+1 and represents 0.
We will only be interested in spectral sequences {(Ep,qr, dr)}r 0 of
bigraded modules with differentials dr of bidegree (r, -r + 1). Such
a spectral sequence strongly converges if for each (p, q) 2 Z x Z there
is a non-negative integer r(p, q) such that dr|Ep,qris the zero homo-
morphism whenever r r(p, q); the definition Ep,q1:= Ep,qris then
independent of r r(p, q) (up to isomorphism) (see [Sp , page 467]).
A spectral sequence as in the previous paragraph is a jth-quadrant
spectral sequence if Ep,qris the trivial module whenever the pair (p, q)
is not in (the closed) quadrant j, j = 1, 2, 3, 4.
A collection of subcomplexes
(3.4) . . .! F pEq-1- ! F pEq -! F pEq+1 ! . . .
of (3.2), indexed by p 2 Z, is a filtration of that complex if {F pEq}p2Z
is a filtration of Eq for each q 2 Z. Any such filtration gives rise to
a spectral sequence of bigraded modules in the following (completely
standard) manner: for each p, q 2 Z and each r 0 define
(3.5) Zp,qr:= { a 2 F pEp+q : da 2 F p+rEp+q+1 } ,
check that dZp-(r-1),q+(r-1)-1r-1 Zp,qr+ F p+1Ep+q, where
dZp-r+1,q+r-2-1:= 0, and set
(3.6) Ep,qr:= (Zp,qr+ F p+1Ep+q)=(dZp-(r-1),q+(r-1)-1r-1+ F p+1Ep+q).
8 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
For fixed r 0 the R-linear mapping d induces R-linear mappings
L
d : Ep,qr! Ep+r,q-r+1r, and the direct sum Er := p,qEp,qris thereby
endowed with the structure of a bigraded module with differential dr
of bidegree (r, -r + 1).
Theorem 3.7. The sequence {(Er, dr)}r 0 is a spectral sequence.
For a proof see, e.g., [Mac , page 346].
Any R-linear mapping f : M ! N between R-modules can be
embedded into the finite complex
f
(3.8) 0 ! M - ! N ! 0 ,
i.e., can be considered as one mapping within the cochain complex
f 1 0
(3.9) . . .! 0 ! 0 ,! E0 := M ! E := N ! 0 ! 0 ! . . ..
When M and N admit filtrations {F pM}p2Z and {F pN}p2Z and
f preserves these filtrations the spectral sequence construction imme-
diately preceding Theorem 3.7 applies (assuming the trivial filtration
on 0). The resulting spectral sequence is the spectral sequence of the
linear (filtration 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 fixed
` 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 identifications so as to relate terms appear-
ing in particular spectral sequence calculations to terms appearing in
normal form calculations. One has
(3.10) Zp,qr= 0 and Ep,qr= 0 if q 6= -p, -p + 1
and
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 9
8 p,-p p -1 p+r
>> (a) Zr = F M \ f (F N),
>>
>> p,-p+1 p
>> (b) Zr = F N,
>> p,-p p+1
>> Zr + F M
>> (c) Ep,-p = ________________
>> r F p+1M
>>
>> F pM \ f -1(F p+rN) + F p+1M
>> = _______________________________
>> p+1
>> F M
< F pM \ f -1(F p+rN)
(3.11) = _______________________, and
>> F p+1M \ f -1(F p+rN)
>> p
>> p,-p+1 F N
>> (d) Er = ______________________________
>> p-(r-1),-p+(r-1) p+1
>> f (Zr-1 ) + F N
>> p
>> F N
>> = ______________________________________
>> f (F p-(r-1)M \ f -1(F pN)) + F p+1N
>>
>> F pN
>> ________________________________
: = f (F p-(r-1)M) \ F pN + F p+1N
In particular,
8 p,-p p
>> (a) Z0 = F M,
>>
>> p,-p+1 p
>> (b) Z0 = F N
>>
>> p p+1
>< (c) Ep,-p F__M_+_F____M___
0 = F p+1M
(3.12)
>> = F pM=F p+1M
>>
>> p
>> p,-p+1 F N
>> (d) E0 = ____________________________
>> f (F p+1M) \ F pN + F p+1N
>:
= F pN=F p+1N .
and
8 p,-p p -1 p+1
>> (a) Z1 = F M \ f (F N),
>>
>> p,-p+1 p
>> (b) Z1 = F N
< p -1 p+1
(3.13) p,-p __F__M_\_f___(F____N)__
>> (c) E1 = p+1 -1 p+1
>> F M \ f (F N)
>> p
>> p,-p+1 F N
: (d) E1 = __________________________.
f (F pM) \ F pN + F p+1N
10 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
When there is an integer k such that the filtrations of the previous
paragraph satisfy F pM = F 0M = M and F pN = F kN = N for all
p < k one checks easily that for any such p and any r 0 one has
8
>> Zp,-pr = M ,
>>
>> p,-p+1
>< Zr = N ,
(3.14) p < k and r 0 ) Ep,-p = M_+_M___ = 0 , and
>> r
>> M
>> N
>: Ep,-p+1r = ___________ = 0 .
f (M) + N
In particular, for k = 0 the spectral sequence of f : M ! N is then
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 differential dr : Ep,-pr! Ep+r,-(p+r)+1ris calculated
by means of elementary linear algebra: one computes the linear map-
ping f |FpM\f-1(Fp+rN) = f |Zp,-prin the top line of the following com-
mutative diagram and interprets the results within the indicated quo-
tients.
(3.15)
f|FpM\f-1(Fp+rN)
Zp,-pr -! F p+rN
= # # =
F pM \ f -1(F p+rN) F p+rN
# #
F_p+rM_\_f_-1(F_p+rN)__ ____________F_p+rN_______________
F p+1M \ f -1(F p+rN) f (F p+1M) \ F p+rN + F p+r+1N
= # # =
Ep,-pr -dr! Ep+r,-(p+r)+1r
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 11
To ease notation express this last diagram as
f p+r
Zp,-pr -! F N
(3.16) oep,r# # op+r,r
Ep,-pr -dr! Ep+r,-(p+r)+1r
and note that both oep,r and op+r,r are epimorphisms (use the equiva-
lences in (3.11)).
Proposition 3.17.
(a) Choose e 2 Zp,-pr F pM and set [e] := oep,r(e) 2 Ep,-pr. Then
the following statements are equivalent.
(i) [e] 2 ker dr;
(ii) [e] survives to Er+1;
(iii)f (e) 2 f (Zp+1,-(p+1)r-1) + F p+r+1N;
(iv) f (e) 2 f (F p+1M) \ F p+rN + F p+r+1N; and
(v) there is an element a 2 Zp+1,-(p+1)r-1such that f (e) - f (a) =
f (e - a) 2 F p+r+1N.
Moreover, if a 2 Zp+1,-(p+1)r-1satisfies the condition in (v) then
e - a represents the class of [e] in Ep,-pr+1.
(b) Suppose ^e2 F p+rN and set [^e] = op+r,r(^e) 2 Ep+r,-(p+r)r. Then
the following statements are equivalent.
(i) [^e] 2 dr(Ep,-pr);
(ii) [^e] is killed by dr; and
(iii)there is an element b 2 Zp,-prsuch that ~e:= f (b) 2 F p+rN
satisfies [^e] = [~e] := op+r,r(~e).
Proof :
(a)
(i) , (ii) : By definition.
(i) , (iii) : By the commutativity of diagram (3.16)
and the initial equality of (3.11d) (with p replaced by
p + r).
(i) , (iv) : Use the final equality of (3.11d).
(iii) , (v) : From the definitions.
12 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
To prove the final assertion first note from F p+1M
F pM that Zp+1,-(p+1)r-1 Zp,-pr F pM, hence a, e 2
F pM, and it follows from (v) that e - a 2 Zp,-pr+1. From
(e - a) - e = -a 2 F p+1M we then see from the first
equality in (3.11c) (with r replaced by r + 1) that e - a
represents the class of [e] in Ep,-pr+1.
(b)
(i) , (ii) : By definition.
(i) , (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 1N, and define f : M ! N by
(i) f : m 2 M 7! ad(`)m = [`, m] 2 N ,
where
0 1
0 0 0 0 4 0 6 7
B 0 0 1 0 0 0 0 12C
B C
B 0 0 0 1 0 3 8 0 C
B 0 0 0 0 1 0 0 0 C
(ii) ` = BB CC .
B 0 0 0 0 0 1 0 0 C
B C
B 0 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., M0 := 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 Ep,-prand Ep,-p+1rfor p 0 and
r 0.
Throughout the calculations we let epk 2 M denote the 8 x 8 matrix
in filtration p with (k, k+p)-entry 1 and all other entries 0, 1 p 7
and 1 k 8 - p. Note that (ep1, . .,.ep,8-p) provides a(n ordered)
basis of Lp. Equivalence classes (cosets) of the epk will be denoted
[epk], regardless of the particular factor space.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 13
The E0 Terms : We have Ep,-p0= F pM=F p+1M ' Mp and Ep,-p+10=
F pN=F p+1N ' Np for all p 0 by (c) and (d) of (3.12).
The E1 Terms : From ` 2 M = F 1N we have
(iii) f (F pM) F p+1N ,
whereupon from (c) and (d) of (3.13) we conclude that Ep,-p1=
F pM=F p+1M = Ep,-p0and Ep,-p+11= F pN=F p+1N = Ep,-p+10. These
isomorphisms would generally be indicated by writing E0 = E1 (or
E**0= E**1).
The diagrams for both the E0 and E1 terms both begin with that
shown below, wherein the notation Epqi for i = 0, 1 is replaced by
nR := R . . .R to indicate a basis dependent vector space isomor-
phism Epq ' Rn 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 E1,-1i' M1 for i = 0 and 1 is ([e11], . .,.[e17]). The
distinction between the two diagrams becomes evident only when the
differentials are added to complete the pictures: for E0 the differential
would be indicated by vertical arrows between nR and nR, and for
E1 by horizontal arrows from nR to (n - 1)R.
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
____________________________________________________________||||||||||||*
*||||||||234567810
| | | | | | | | | |
0| @ r | @7r|R | | | | | | |
____________________________________________________________||||||||||||*
*||||||||@@
@ @
-1 || ||@ _______-||||@r||r7R6||R || || || ||
____________________________________________________________||||||||||||*
*||||||||@@
@ @
-2 || || ||@ _______-||||@r||r6R5||R || || ||
____________________________________________________________||||||||||||*
*||||||||@@
@ @
-3 || || || ||@ _______-||||@r||r5R4||R || ||
____________________________________________________________||||||||||||*
*||||||||@@
@ @
-4 || || || || ||@_______-||||@rr||4R3R|| ||
____________________________________________________________||||||||||||*
*||||||||@@
@ @
-5 || || || || || ||@_______-||||@rr||3R2R||
____________________________________________________________||||||||||@@
| | | | | | @| |@ | |
-6 || || || || || || ||@_______-||||@rr||2R1R
____________________________________________________________||||||||||@@
| | | | | | | @| |@ |
| | | | | | | |@ | @ |
-7 | | | | | | | | | |
____________________________________________________________||||||||||
The E0 and E1, d1 terms.
14 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
The E2 Terms : This requires calculating the mappings d1 : Ep,-p1!
Ep+1,-p1, and we do so as in (3.16) (more precisely, as in (3.15))
with r = 1. The condition `0 = 0 gives Zp,-p1= F pN = F pM
for p 1, and as a consequence it suffices to calculate the effect of
ad (`)|FpN : F pN ! F p+1N on the basis elements epj and then pass
to quotients. The calculations are completely straightforward, and the
results are summarized in the following table, wherein the initial entry
p = 1, [e11] 7! -[e21] indicates that d1 : [e11] 2 E1,-117! -[e21] 2
E2,-11, etc.
[e11] ! -[e21]
[e21] ! -[e31]
[e12] ! -[e22]
[e22] ! -[e32]
[e13] ! [e22]- [e23]
[e23] ! [e32]- [e33]
p = 1 [e14] ! [e23]- [e24] p = 2
[e24] ! [e33]
[e15] ! [e24]
[e25] ! [e34]
[e16] ! [e25]
[e26] ! [e35]
[e17] ! 0
[e31] ! -[e41]
[e41] ! -[e51]
[e32] ! -[e42]
[e42] ! 0
p = 3 [e33] ! [e42] p = 4
[e43] ! [e52]
[e34] ! [e43]
[e44] ! [e53]
[e35] ! [e44]
[e51] ! 0
[e61] ! 0
p = 5 [e52] ! 0 p = 6
[e62] ! 0
[e53] ! [e62]
We can use these calculations to illustrate the spectral sequence
jargon introduced earlier: [e17] and [e12] + [e13] + [e14] + [e15] 2 E1
survive to E2, [e11] 2 E1 does not, and [e31] 2 E3,-21is killed (by
-[e21] 2 E2,-21), as is [e42] (by -[e32]). In particular, [e31] and [e42]
must survive to E2 and represent 0 .
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 15
From the calculations above the cohomology E2 of E1, described in
terms of associated generators (i.e., basis elements), is easily seen to be
E1,-12: [e17] a = [e12 + e13 + e14 + e15]
E2,-12: [e26]
E2,-22: b = [e22 + e23 + e24]
E3,-32: c := [e32 + e33] ,
E4,-42: [e42]
E5,-52: [e51] [e52]
E6,-52: [e61]
and the associated|diagram|for|E2|is|therefore||||||||||| || ||
| | | | | | | | | |
____________________________________________________________|||||||||||*
*|||||||||234567810
| | | | | | | | | |
0 | @ r | @7r|R | | | | | | |
____________________________________________________________|||||||||||*
*|||||||||@@
| |@@ r2|@R@ r|1R | | | | | |
-1 | | H H| | | | | | | |
________________H_H_________________________________________|||||||||||*
*|||||||||@@
| | |@ H|Hj@ | | | | | |
-2 | | | @ HrH|1@rR|0 | | | | |
______________________HH____________________________________|||||||||||*
*|||||||||@@
| | | |@ H|Hj@ | | | | |
-3 | | | | @HrH|1@Rr0| | | | |
___________________________H_H______________________________|||||||||||*
*|||||||||@@
| | | | @| H|Hj@ | | | |
-4 | | | | |@ HrH|1@Rr0| | | |
_________________________________H_H________________________|||||||||||*
*|||||||||@@
| | | | | @| H|Hj@ | | |
-5 | | | | | |@ HrH|2@Rr1|R | |
_______________________________________H_H__________________|||||||||||*
*|||||||||@@
| | | | | | @| H|Hj@ | |
-6 | | | | | | |@ r2 |R@ r1|R |
____________________________________________________________||||||||||@@
| | | | | | | @| |@ |
| | | | | | | |@ | @ |
-7 | | | | | | | | | |
____________________________________________________________||||||||||
The E2, d2 term.
wherein 0 denotes the trivial module. (Recall that unlabeled vertices
also represent the trivial module.)
The E3 Terms : We need to compute d2 : Ep,-p2! Ep+2,-p-12. From
the last diagram we see that only possible nontrivial components of
this homomorphism arise in the contexts E4,-42 R ! E6,-52 R and
E5,-52 2R ! E7,-62 R.
Applying (3.16) with r = 2 we obtain the following analogue of the
first collection of displayed formulas within the discussion of the E2
16 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
terms:
[e51] ! -2[e71]
p = 4 [e42] ! -2[e62] p = 5
[e52] ! 0
The diagram for the E3 terms appearing below is an easy consequence.
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
____________________________________________________________||||||||||||*
*||||||||234567810
| | | | | | | | | |
0| @ r | @7r|R | | | | | | |
____________________________________________________________||||||||||||*
*||||||||@@
| |@ |@ | | | | | | |
-1 | | @ r2|R @ r|1R | | | | | |
____________________________________________________________||||||||||||*
*||||||||@@
| | |@ |@ | | | | | |
-2 | | | @ r1|R @ r|0 | | | | |
____________________________________________________________||||||||||||*
*||||||||@@
| | | |@@ r1|@@Rr0| | | | |
-3_______________________Q_Q__________________________________||||||||||||*
*||||||||@@
|| || || || |Q|Q@ ||@ || || || ||
-4 | | | | |@ r1Q|Q@Rr0| | | |
__________________________________Q_Q_______________________||||||||||||*
*||||||||@@@@
-5 || || || || || ||@r1 |Qs|@r||R1R || ||
____________________________________________________________||||||||||@@
| | | | | | @| |@ | |
| | | | | | |@ r2 |R@ r0| |
-6 | | | | | | | | | |
____________________________________________________________||||||||||@@
| | | | | | | @| |@ |
| | | | | | | |@ | @ |
-7 | | | | | | | | | |
____________________________________________________________||||||||||
The E3, d3 term.
This seems an appropriate place to ease the formality of our presen-
tation: in practice the observations resulting in the E3 diagram would
more likely be stated along the following lines.
The space E4,-42is generated by [e42], which is mapped
by d2 to -2[e62] = 0 2 E6,-52([e62] was killed by [e53]).
The mapping d2 : E4,-42! E6,-52is therefore the zero
transformation, and as a consequence [e42] survives to
E3. The class [e42] is represented in E3 by [e42 + 2e53].
The space E5,-52is generated by [e51] and [e52], and one
checks that d2([e51]) = -2[e71] and d2([e52]) = 0.
The E4 Terms : The only possible nontrivial (component of) d3 :
Ep,-p3! Ep+3,-p+23is (the restriction to) E3,-33! E6,-53. However, one
checks that E3,-33is generated by [e32+ e33], and that d3 carries this
class to 0. E4 = E3 follows.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 17
The E5 Terms : The only possible nontrivial d4 is E2,-24! E6,-54.
The first of these spaces in generated by [b], and d4([b]) = 0. E5 = E4
follows.
The E6 Terms : The only possible nontrivial d5 is E1,-15! E6,-55.
The first of these spaces in generated by [e17] and [a], and d5 anni-
hilates both. E6 = E5 follows.
The E7 Terms : The differential d6 is trivial, hence E7 = E6.
The E1 Terms : All dr with r 6 and trivial, hence E1 = E6 =
E3 (in the sense that Ep,q1= Ep,q3for all (p, q) 2 Z x Z). There is a
single generator !2 for E2,-11and a single generator !6 for E6,-51; all
the other vector spaces Ep,-p+11are trivial.
We have calculated the spectral sequence of f = ad (`) : M ! N
completely, and in the process have established strong convergence.
18 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
4. A Brief Introduction to Normal Form Theory
L
Throughout this section L = Ls denotes a Z-graded R-Lie al-
gebra with Ls = 0 if s < 0 and sss : L ! Ls is used to denote
the associated projections. We write the typical element of Ls as `s
and view each Ls as a subspace of L by means of the obvious sec-
tion Ls ! L, i.e., we identify an element `s 2 Ls with the element
. . .+ 0 + `s + 0 + . . .2 L when confusion cannot otherwise result.
Suppression of notational reference to the sections Ls ! L is a com-
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 fix an element `0 2 L0. We do not exclude
the choice `0 = 0.
Definition 4.1. An element ` = `0 + `1 + . .+.`s + . . .2 L is in
classical 1normal form to order s 0 if `j 2 ker(ad (`0)|Lj) for j =
0, . .,.s, and is in classical normal form if this is the case for all s 0.
In other words, ` is in normal form (to order s) if [`0, `j] = 0 for
all 0 j ( s). Note from [`0, `0] = 0 that ` is always in classical
normal form to order 0.
An element `0 2 L0 splits L if
(4.2) Lj = ker(ad (`0)|Lj) im(ad (`0)|Lj), j 1 .
Proposition 4.3. (The Classical Normal Form Algorithm) Sup-
pose `0 splits L and ` = `0 + . .+.`s + . . .is in classical normal
form to order s. Write `s+1 = `Ks+1+ `Is+1 in accordance with the
decomposition (4.2) with j = s + 1. Choose m 2 Ls+1 such that
ad (`0)m = [`0, m] = `Is+1. Then ` + ad(m)` is in classical normal form
to order s + 1 and Js(` + ad(m)`) = Js(`).
This formulation is adapted from [CKR ], but did not originate therein.
Proof : This is evident from the following calculation, where in each
Q
line the final dots represent terms in t s+2Lt.
____________
1The "classical" designation is not standard: it has been added to distingui*
*sh
these normal forms from those introduced later.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 19
` + ad(m)` = Js(`) + `Ks+1+ `Is+1+ . .+.[m, `0 + . .+.`s+1 + . .].
= Js(`) + `Ks+1+ `Is+1+ [m, `0] + . . .
= Js(`) + `Ks+1+ `Is+1- [`0, m] + . . .
= Js(`) + `Ks+1+ `Is+1- `Is+1+ . . .
= Js(`) + `Ks+1+ . ...
q.e.d.
For m 2 L define ad0(m) := idL : L ! L, and if i 1 and
ad i-1(m) : L ! L has been defined set ad i(m) := ad(m) O adi-1(m) :
L ! L.
To see how the algorithm can be applied in practice assume, for the
remainder of the section, that R is a field of characteristic 0. Then
for any m 2 F 1L a linear mapping expad : L ! L is defined by
X1
1 i
(4.4) expad (m) := __ad (m).
i=0 i!
Indeed, by (2.4) and the assumption m 2 F 1L the formal expression
(4.5) expad (m)` = ` + [m, `] + 1_2[m, [m, `]] + . . .
involves only finite sums in each Lp, and therefore represents a well-
defined element of L. In fact expad (m) : L ! L is a K-Lie algebra
automorphism2, i.e.,
(4.6)
expad (m)[`, ^`] = [expad (`), expad (^`)], m 2 F 1L, `, ^`2 L .
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 1L and B 2 L. Then one sees by writing out the
Taylor series for f (t) = eMt Be-Mt at t = 0 and evaluating at t = 1
that
(i) expad (M)B = eM Be-M .
The next proposition shows that the adjoint representation in algo-
rithm (4.3) may be replaced with expad .
____________
2When 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 `0 splits L and ` = `0 + . .+.`s + . . .
is in classical normal form to order s. Write `s+1 = `Ks+1+ `Is+1in
accordance with the decomposition (4.2) with j = s + 1. Choose m 2
Ls+1 such that ad(`0)m = [`0, m] = `Is+1. Then expad (m)` is in
classical normal form to order s + 1 and Js(expad (m)`) = Js(`).
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-1 , . .,.M1 2
F 1T such that conjugating A by the product eMn . . .eM1 converts A
to the appropriate classical normal form.
Group actions enter the picture by first noting that the graded vector
subspace G := F 1L L is a filtered group w.r.t. the binary operation
* defined by the Campbell-Hausdorff formula
(4.10) m * n = m + n + 1_2[m, n] + 1_12[m, [m, n]] + . . .
(e.g., see [BC ] and/or [Se , 14.15]) : the filtration {F pG}p 1 of G is
Q
defined by the inherited grading, i.e., F pG := q 1 Gq, where Gq := Lq
for all q 1; the identity element is 0; the inverse of m 2 G is -m.
Definition (4.10) is designed so as achieve
(4.11) expad (m * n) = expad (m) expad (n), m, n 2 G ,
where expad (m) expad (n) := expad (m) O expad (n), and it follows that
the mapping (m, `) 2 G x L ! expad (m)` defines 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 con-
struction of an orbit representative of `, although for the actual exis-
tence proof one needs to establish convergence in the filtration topology
of G.
There are two significant problems with the classical theory:
o classical normal forms obtained by successive applications of Propo-
sition 4.8 are generally not unique; and
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 21
o when `0 2 L0 does not split L there is no algorithm to guarantee
that one can always convert an element ` = `0 + `1 + . .2.L to
classical normal form.
The first problem was generally treated by attempting "further refine-
ments" of elements in classical normal form; the second by replacing
ker(ad (`0)|Lj) in (4.2) with a suitable complement of im(ad (`0)|Lj)
(often associated with the representation theory of sl(2, C)). Of course
each of these approaches required modifications of Definition 4.1. A.
Baider [B ] gave an elegant solution to both problems by replacing
im (ad (`0)|Lj) in the decomposition of Lj with a generally larger sub-
space and assuming a prescribed complement, e.g., the orthogonal com-
plement w.r.t. a given inner product on Lj.
To describe Baider's method assume ` = `0 + `1 + . .2.L has been
given, define3
(4.12)
C1s(`) := { m 2 G := F 1L = F 1G : [m, `] 2 F s+1L }, s 0,
and then define
1
(4.13) Vs1+1(`) := sss+1 ad (`)(Cs(`)) Ls+1, s 0.
Note that when s 0 and m 2 C1s(`) one has
(4.14) [expad (m)`] = [` + [m, `]] 2 L=F s+1L ;
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 Vs1(`) has been chosen
which depends only on Js-1(`), hence that
(4.15) Ls = Ys(`) Vs1(`), s 1 .
In particular,
(4.16) Vs1(`) = Ls , Ys(`) = 0.
To involve all non-negative indices in the definition of Vs1(`) define
(4.17) Y0(`) := L0.
____________
3Baider refers to the Lie subalgebra C1s(`) L as the s-order "centralizer"*
* of
`, and employs slightly different notation. Our notation is designed to make the
connection with spectral sequences more transparent.
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 ].
Definition 4.18. An element ` = `0 + `1 + . .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.
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 ` = `0 + . .+.`s + . . .is in normal form
to order s 0. Write `s+1 = `Ys+1+ `Vs+1 in accordance with the
decomposition (4.15) (with s replaced by s + 1). Choose m 2 C1s(`)
such that sss+1 ad (`)m = `Vs+1. Then expad (m)` is in normal form
to order s + 1 and Js(expad (m)`) = Js(`).
Proof : Immediate from Proposition 4.3, (4.14), and the assumption
that Ys+1(`) depends only on Js(`). q.e.d.
We can now be more explicit about one of the goals of the paper:
we will show, in somewhat greater generality, that the calculations in-
volved in applying Proposition 4.19 to specific normal form problems
are simply special cases of spectral sequence calculations as in Exam-
ple 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 defined by Proposition 4.19 is used to produce elements
ms 2 G to convert expad (ms-1 * . .*.m1)` to normal form of order
s + 1 the sequence {ms * . .*.m1} converges in G to an element m
such that expad (m)` = `N .
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 C10(`) L0 and V11(`) L1 is
always straightforward. Specifically, one sees from the definition that
C10(`) = F 1L = G, and from (2.4) that V11(`) := ss1 ad (`)(C10(`)) =
ss1 ad (`)(G) = ss1 ad (`0 + . . .)(L1 F 2G) = ss1 ad (`0)(L1)
= ad(`0)(L1), where in writing ss1 ad (`0)(L1) we are identifying
ad (`0)(L1) with its image in L under the obvious section L1 ! L
of ss1. In summary:
(4.21) C10(`) = G and V11(`) = ad(`0)(L1).
In special cases the calculation of Vs1+1(`) is also easy: for any s 0
one has
(4.22) Ls C1s(`)
(more precisely: Ls = sss(C1s(`))), hence
(4.23) ad(`0)(Ls+1) Vs1+1(`) ,
and it follows that
(4.24)
ad (`0)(Ls+1) = Ls+1 ) Vs1+1(`) = Ls+1 and Ys+1(`) = 0
(recall (4.16)).
Other easy cases arise. For example, when `0 = 0 one sees from
(4.12) that C11(`) = G, whence from (2.4) that V21(`) = ss2 ad (`)(G)
= ad(`1)L1, i.e.,
(4.25) `0 = 0 ) V21(`) = ad(`1)L1 .
Unfortunately, the determination of C1s(`) and (thence) Vs1+1(`) can
in general be a daunting task, although it is difficult to appreciate this
assertion until one begins working with specific examples. (With the
spectral sequence approach the calculation of Vs1+1(`) becomes com-
pletely 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+1 2 Vs1+1(`) without complete knowledge
of either C1s(`) or Vs1+1(`), in which case it is clear from the normal
form algorithm that the normal form `N must satisfy `Ns+1= 0.
An obvious approach to computing C1s(`) is to work upward through
the filtration
(4.26) Cs1(`) Cs-12(`) . . .C1s(`)
24 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
of Lie subalgebras defined by
(4.27)
Cps-p+1(`) := { m 2 F pG : ad (`)m 2 F s+1L }, p = s, s - 1, . .,.1,
and with this in mind we define the initial terms Ips-p+1(`) of Cps-p+1(`)
by
p
(4.28) Ips-p+1(`) = ssp Cs-p+1(`) Lp, p = s, s - 1, . .,.1.
The initial terms of Cs1(`) are easy to compute: we claim that
(4.29) Is1(`) = ker ad (`0)|Ls .
Indeed, for m = ms + ms+1 + . .2.F sL see from (2.4) that
ad (`)m = [`0 + `1 + . .,.ms + ms+1 + . .].
= [`0, ms]s + {terms in F s+1L},
and the claim follows. As a consequence of (4.29) and (4.26) we see
that
s
(4.30) `0 = 0 and s 1 ) Is1(`) = sss C1(`) = Ls.
However, from the definitions (and the subspace identification conven-
tions) one sees that Is1(`) Cs1(`), and it follows that
(4.31)
`0 = 0 and s 1 ) Ls Cs1(`) and ad(`1)Ls Vs1+1(`).
We need a practical characterization of the initial terms of Cps-p+1(`).
Suppose 1 p < s and mp 2 Lp. Then mp completes in Cps-p+1(`)
if there is an element m^ 2 F p+1G such that mp + ^m2 Cps-p+1(`).
Proposition 4.32. For any 1 p < s and any mp 2 Lp the follow-
ing statements are equivalent:
(a) mp 2 Ips-p+1(`), i.e., mp is an initial term of Cps-p+1(`);
(b) the element mp completes in Cps-p+1(`);
(c) one has
ad (`)(mp) 2 ad(`)(F p+1L) + F s+1L ;
and
(d) one has
p+1 s+1
[0] = [ad (`)(mp)] 2 F pL= ad(`)(F L) + F L .
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 25
Proof : For m^ 2 F p+1L we have
mp + ^m2 Cps-p+1(`) , ad(`)(mp + ^m) = 0 mod F s+1L
, ad(`)mp = ad(`)(-m^) mod F s+1L ,
and the equivalences follow. q.e.d.
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 Ips-p+1(`) , ad(`)(mp) 2 ad(`)(F p+1L) + F s+1L .
Example 4.34. We offer 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 `0 does not split L), and we have therefore chosen to con-
sider 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 exam-
ple. The splitting convention is that defined by the inner product
:= tr(AoB) on L, i.e., in the direct sum decompositions (4.15)
we take Yp(`) := Vp1(`)? Lp .
We compute the normal form of the nilpotent matrix
0 1
0 0 0 0 4 0 6 7
B 0 0 1 0 0 0 0 12C
B C
B 0 0 0 1 0 3 8 0 C
B 0 0 0 0 1 0 0 0 C
(i) ` := BB CC
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
appearing (not coincidentally) in Example 3.18, and to use the methods
introduced we write ` in the form `0+`1+. .+.`7, wherein `0 denotes
26 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
the zero matrix,
0 1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7
B 0 0 1 0 0 0 0 0C B 0 0 0 0 0 0 0 0C
B C B C
B 0 0 0 1 0 0 0 0C B 0 0 0 0 0 0 0 0C
B 0 0 0 0 1 0 0 0C B 0 0 0 0 0 0 0 0C
`1 = BB CC, . .,.`7 = BB CC.
B 0 0 0 0 0 1 0 0C B 0 0 0 0 0 0 0 0C
B C B C
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A @ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The normal form of ` to order s 0 is written `(s)= `(s)0+ `(s)1+
`(s)2+ . . ..
Order 0 : As noted immediately following Definition 4.18, the element
` is automatically in normal form to order 0, hence `(0)= `.
Order 1 : Since `(0)0= `0 = 0 we see from (4.21) that V11(`(0)) = 0,
hence Y1(`(0)) = L1, and we conclude that `(0) is also in normal form
to order 1. It follows from the uniqueness of normal forms that `(1)=
`(0)= `. In the notation of Remark 4.9 we take M1 to be the zero
matrix, and eM1 is then the identity matrix I = I8.
Order 2 : By (4.25) we have V21(`(1)) = ad (`(1)1)(L1) = ad (`1)(L1),
and by elementary calculation one verifies that this last subspace of
L2 consists of those elements mij 2 L2 with m68 = 0. From the
definition Y2(`(1)) = V21(`(1))? we conclude that V21(`(1)) consists of
those elements mij2 L2 in which all entries other than m68 must be
zero, hence `(1)22 Y2(`1), and `(2)= `(1)= `(0)= ` follows. We take
M2 as the zero matrix, resulting in eM2 = I.
Order 3 : Check that the matrix M2 2 L2 with 3 in the (4, 6)
position and zeros elsewhere satisfies ad(`(2))(M2) = `(2)3. It follows
from (4.31) that `(2)32 V31(`(2)), hence that `(3)3= 0. To calculate
`(3) completely note that eM2 = I + M2 ; then check that
0 1
0 0 0 0 4 0 6 7
B 0 0 1 0 0 0 0 12C
B C
B 0 0 0 1 0 0 8 0 C
B 0 0 0 0 1 0 0 6 C
`(3)= expad (M2)`(2)= eM2 `(2)e-M2 = BB CC .
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
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 ff 2 R the matrix
0 1
0 0 0 -4 0 0 0 0
B 0 0 0 0 ff 0 0 0C
B C
B 0 0 0 0 0 ff 0 0C
B 0 0 0 0 0 0 8 0C
M3 = BB CC 2 L3 C31(`(3)) C22(`(3)) C13(`(3))
B 0 0 0 0 0 0 0 6C
B C
B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
satisfies ad (`(3))(M3) = `(3)4, hence `(4)4= 0. One has
0 1
1 0 0 -4 0 0 -16 0
B 0 1 0 0 ff 0 0 3ffC
B C
B 0 0 1 0 0 ff 0 0 C
B 0 0 0 1 0 0 8 0 C
eM3 = BB CC,
B 0 0 0 0 1 0 0 6 C
B C
B 0 0 0 0 0 1 0 0 C
@ 0 0 0 0 0 0 1 0 A
0 0 0 0 0 0 0 1
from which one obtains
0 1
0 0 0 0 0 0 6 -17
B 0 0 1 0 0 0 0 12 C
B C
B 0 0 0 1 0 0 0 2ffC
B 0 0 0 0 1 0 0 0 C
`(4)= expad (M4)`(3)= BB CC.
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
This illustrates non-uniqueness within the Mj.
Order 5 : We make the choice ff = 0 in the previous step; the matrix
`(4) is then seen to be in normal form to order 5, hence `(5)= `(4).
(By the uniqueness of normal forms any other choice for ff would have
[ultimately] resulted in an `(5) with the same 5-jet.) We take M4 = 0,
hence eM4 = I.
Order 6 : Here the method used for Orders 3 and 4 fails: one easily
verifies that `(6)2= ad(`(5))(L5), and as a result we cannot appeal to
(4.31) to conclude that `(5)52 V61(`(5)). This is the first case in which
Proportion 4.32, in the guise of (4.33), plays a significant role. We ex-
p
amine the initial terms Ip5-p+1(`(5)) := ssp C5-p+1(`(5)) as p decreases
28 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
from 4, recalling from (4.33) that
(ii) mp 2 Ip5-p+1(`(5)) , ad (`(5))mp 2 ad(`(5))(F p+1L) + F 6L .
We offer a somewhat detailed presentation of this case so as to empha-
size the completely elementary nature of the calculations.
The Initial Terms I42(`(5)) = ss4 C42(`(5)) : In this case (ii)
becomes
(iii) m4 2 I42(`(5)) , ad (`(5))m4 2 ad(`(5))(F 5L) + F 6L .
However, from `(5)0= `0 = 0 and (2.4) we see that ad(`(5))(F 5L)
F 6L, whereupon (iii) reduces to
(iv) m4 2 Ip2(`(5)) , ad (`(5))m4 2 F 6L .
The Lie subalgebra F 6L L consists of all matrices of the form
0 1
0 0 0 0 0 0 * *
B 0 0 0 0 0 0 0 * C
B C
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
(v) B C ,
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
and for a typical element
0 1
0 0 0 0 m15 0 0 0
B 0 0 0 0 0 m26 0 0 C
B C
B 0 0 0 0 0 0 m37 0 C
B 0 0 0 0 0 0 0 m C
(vi) m4 := BB 48 CC2 L4
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
we have
0 1
0 0 0 0 0 -m15 0 0
B 0 0 0 0 0 0 m37 -2m26 C
B C
B 0 0 0 0 0 0 0 m48 C
B 0 0 0 0 0 0 0 0 C
(vii) ad(`(5))(m4) = BB CC.
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 29
It follows immediately from (vii) that I42(`(5)) = ss4 C42(`(5)) consists
of those m4 as in (vi) with m15 = m37 = m48 = 0, i.e., that a matrix
m4 2 L4 completes in C42(`(5)) if and only if this matrix has the form
0 1
0 0 0 0 0 0 0 0
B 0 0 0 0 0 m26 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
B 0 0 0 0 0 0 0 0 C
(viii) m4 := BB CC.
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
Now let m = m4 + ^m 2 I42(`(5)) + F 6L be an arbitrary element of
C42(`(5)). Then ad(`(5))m^ 2 F 7L, and we conclude from (vii) that
ad (`(5))(C42(`(5))) has the form seen in (v) when the (1, 7)-entry has
been replaced by 0. Since the (1, 7)-entry 6 of `(5)(= `(4)) is not zero
this means that more work is required to determine if `(5)2 V61(`(5)).
We therefore ascend to C33(`(5)).
The Initial Terms I33(`(5)) = ss3 C33(`(5)) : Here (ii) becomes
(ix) m3 2 I33(`(5)) , ad (`(5))m3 2 ad(`(5))F 4` + F 6`.
One checks that ad(`(5))F 4L + F 6L = F 5L, and (ix) is thereby re-
duced to
m3 2 I33(`(5)) , ad (`(5))m3 2 F 5L.
Now check that for the typical element
0 1
0 0 0 m14 0 0 0 0
B 0 0 0 0 m25 0 0 0 C
B C
B 0 0 0 0 0 m36 0 0 C
B 0 0 0 0 0 0 m 0 C
m3 := BB 47 CC 2 L3
B 0 0 0 0 0 0 0 m58 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
30 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
we have
0 1
0 0 0 0 -m14 0 0 0
B 0 0 0 0 0 m36 - m25 0 0 C
B C
B 0 0 0 0 0 0 m47 -2m36 C
B 0 0 0 0 0 0 0 m C
ad (`(5))m3 := BB 58 CC ,
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
and as a result we see that I33(`(5)) consists of those matrices of L3 of
the form
0 1
0 0 0 0 0 0 0 0
B 0 0 0 0 m25 0 0 0C
B C
B 0 0 0 0 0 m25 0 0C
B 0 0 0 0 0 0 0 0C
m3 := BB CC ,
B 0 0 0 0 0 0 0 0C
B C
B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
and the typical element of C33(`(5)) has the form
0 1
0 0 0 0 m15 m16 m17 m18
B 0 0 0 0 m25 m26 m27 m28 C
B C
B 0 0 0 0 0 m25 m37 m38 C
B 0 0 0 0 0 0 0 m C
m3 + ^m= BB 48 CC.
B 0 0 0 0 0 0 0 0 C
B C
B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0
By direct calculation one checks that
0 1
0 0 0 0 0 * 0 *
B 0 0 0 0 0 0 * *C
B C
B 0 0 0 0 0 0 0 *C
B 0 0 0 0 0 0 0 0C
ad (`(5))(m3 + ^m) = BB CC ,
B 0 0 0 0 0 0 0 0C
B C
B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
and we immediately conclude, as in the final assertion of the previous
case, that additional work is needed to determine if `(5)62 V51(`(5)).
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 31
The remaining initial terms relating to the order 6 calculation, i.e.,
I24(`(5)) and I15(`(5)), are handled analogously, and in both cases one
finds that the typical matrices in ad(`(5))(Cj6-j(`(5))) 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 V61(`(5))
consists of those matrices as in (ii) with the upper-right entry replaced
Y (5)V61
by 0. The splitting `(5)6= `(5)6+` of Proposition 4.19 is therefore
given by
0 1 0 1
0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 12C
B C B C
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 0 C
B C B C
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 0 C
B C + B C ,
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 0 C
B C B C
B 0 0 0 0 0 0 0 0C B 0 0 0 0 0 0 0 0 C
@ 0 0 0 0 0 0 0 0A @ 0 0 0 0 0 0 0 0 A
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
and from (iv) we see that the matrix
0 1
0 0 0 0 0 0 0 0
B 0 0 0 0 0 -6 0 0C
B C
B 0 0 0 0 0 0 0 0C
B 0 0 0 0 0 0 0 0C
M5 := BB CC 2 I42(`(5)) C42(`(5))
B 0 0 0 0 0 0 0 0C
B C
B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
V61
satisfies ad (`(5))M5 = `(5) . One has expad (M5) = I + M5, hence
0 1
0 0 0 0 0 0 6 -17
B 0 0 1 0 0 0 0 0 C
B C
B 0 0 0 1 0 0 0 0 C
B 0 0 0 0 1 0 0 0 C
`(6)= eM5 `(5)e-M5 = BB CC.
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
32 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Order 7 : The calculation of `(7) involves no new ideas: suffice it to
note that for
0 1
0 0 0 0 0 17=2 0 0
B 0 0 0 0 0 0 0 0C
B C
B 0 0 0 0 0 0 0 0C
B 0 0 0 0 0 0 0 0C
M6 := BB CC 2 I52(`(6))
B 0 0 0 0 0 0 0 0C
B C
B 0 0 0 0 0 0 0 0C
@ 0 0 0 0 0 0 0 0A
0 0 0 0 0 0 0 0
one has eM6 = I + M6 and
0 1
0 0 0 0 0 0 6 0
B 0 0 1 0 0 0 0 0C
B C
B 0 0 0 1 0 0 0 0C
B 0 0 0 0 1 0 0 0C
`(7)= eM6 `(6)e-M6 = BB CC.
B 0 0 0 0 0 1 0 0C
B C
B 0 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
0 1
1 0 0 -4 0 -7=2 -16 0
B 0 1 0 0 0 -6 0 0C
B C
B 0 0 1 0 0 0 0 0C
B 0 0 0 1 0 3 8 0C
eM7 eM6 . .e.M1= BB CC.
B 0 0 0 0 1 0 0 6C
B C
B 0 0 0 0 0 1 0 0C
@ 0 0 0 0 0 0 1 0A
0 0 0 0 0 0 0 1
The splitting convention in the previous example was defined by
an inner product on the graded Lie algebra L. We denote such a
graded Lie algebra by {L, [-, -], <-, ->}. We shall always assume that
the splitting convention specified by {L, [-, -], <-, ->} is given by
orthogonal complements with respect to <-, ->. 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, {L, [-, -], <-, ->}. Then an element ` = `0 +
`1 + . . .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+1 + . . .then mp is perpendicular to `p. Furthermore each
orbit of the action of G = F 1L contains a unique representative in
normal form.
Proof :
) When [g, `] = mp + mp+1 + . . .we have g 2 C1p(`), hence mp 2
Vp1(`) = Yp(`)? . But ` in normal form means `p 2 Yp(`), and the
asserted condition follows.
( For any mp 2 Vp1(`) Lp there is (by definition) an element
g 2 G such that [g, `] = mp + . ... The given hypothesis therefore
implies `p Vp1(`)? = Yp(`), and we conclude that ` is in normal
form (to order s).
Existence and uniqueness was established in Theorem 4.20.
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 filtrations {F pM} and {F pN}, and cosets of submodules
are indicated with brackets. We assume that F 1M is a group w.r.t.
a binary operation * possibly distinct from + , and we define G :=
(F 1M, *). We assume in addition that F pG := F pM for p 1 defines
a filtration of G by subgroups.
For our purposes the appropriate general setting for the normal form
algorithm is an action of a filtered group G on a filtered vector space
having the property that the representation of each element g 2 G is
"linear modulo higher filtrations". Here we make this idea precise.
Definition 5.1. A (set-theoretic) mapping f : M ! N is initially
linear if it preserves the filtrations, i.e.,
(5.2) f (F pM) F pN for all p 2 Z,
and has the form
(5.3) f = fL + fH ,
were fL, fH : M ! N also preserve the filtrations, fL is R-linear, and
for each (m, p) 2 M x Z the following condition holds:
(5.4) 0 = [fL(m)] 2 N=F pN ) 0 = [fH (m)] 2 N=F p+1N .
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 fixed decomposition is always assumed.
For the remainder of the section we let ' : (g, n) 2 GxN 7! g.n 2 N
denote a filtration-preserving left action of G on N, i.e., an action such
that
(5.5) F iG . F jN F i+jN for all (i, j) 2 Z+ x Z.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 35
Definition 5.6. We say that the action ' : G x N ! N is ini-
tially linear if for each ` 2 F 0N the mapping f `: G ! N defined by
f `: g 7! g . ` - ` is initially linear.
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 0L the mapping ad(`) : L ! L is linear, hence
initially linear with ad (`)L = ad(`).
(b) Assuming R is a field of characteristic zero let G := F 1L with
the Campbell-Hausdorff 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 field of characteristic zero and let n be a positive
integer. Then the collection gl(n, R) of n x 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
|i| n.
Take L := T (n), where T (n) gl(n, R) is as in Example
2.5(a). Then G := F 1L 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 1L is invariant under
this action there is an induced action of G on the quotient (vector)
space N := gl(n, R)=F 1L. This quotient is not a R-Lie algebra,
since F 1L is not a Lie ideal of gl(n, R), but it does inherit a
Z-grading via Ni := ss(gl(n, R)i-(2n-1)) for n i 2n - 1.
The shift in indexing is to satisfy the filtration conditions in the
definition of an initially linear group action. One checks easily
that the action of G on N is filtration-preserving.
The quotient space N can obviously be identified with the
Lie subalgebra TL(n) gl(n, R) consisting of lower triangular
matrices (with non-zero diagonal elements allowed), and the in-
duced action of G can then be described as follows: for M 2 G
and N 2 N ' TL(n) we have M . N := ss(eM Ne-M ), where
ss : gl(n, R) ! TL(n) replaces all entries above the diagonal of a
36 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
given matrix with zeros. Equivalently:
M . N := ss N + [M, N] + 1_2![M, [M, N]] + . . ..
It is a simple matter to check that action (M, N) 2 G x N 7!
M . N 2 N is initially linear if we take
fLN : M 2 G 7! (ss O -ad (N))M 2 N .
(d) Take R = C, define Lp = gl(n, C) . zp for all p 2 Z and
Q
set L := [p2Z q pLq. Define the bracket of Azp 2 Lp and
Bzq 2 Lq by
[Azp, Bzq] = [A, B]zp+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-pz-p + . .+.A-1z-1 + A0 + A1z + . . .
in (the complex variable) z with coefficients in gl(n, C).
Set G := F 1L, with the Campbell-Hausdorff group structure.
Define an action of G on L by g . ` = expad (-g)` + d_dzg. (The
derivative represents formal term-by-term differentiation of a se-
ries). This action is initially linear with fL`: m ! [`, m] + _d_dzm
provided one appropriately modifies the definition of "initially lin-
ear action" to take into account the negatively graded terms. We
will not peruse this here.
This example arises when normalizing a first order system y 0=
A(z)y of meromorphic ordinary differential equations on C at a
singularity, w.l.o.g. 0. Specifically, the substitution y = P -1w =
(P (z))-1w converts this equation to w 0= (P A(z)P -1+P 0P -1)w,
and one checks that (P, A(z)) 7!
P A(z)P -1+P 0P -1 defines a left action of Gl (n, C((z))) on gl(n, C((z)*
*)),
where C((z)) is the quotient field of the formal power series ring
C[[z]]. This is the action by gauge transformations. To achieve
our context take P = eg.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 37
(e) An R-Lie algebra, M = (M, [ , ]) is cyclically graded (of order
L t-1
t) if M is the internal direct sum j=0 Mj of R-subspaces
satisfying
[Mp, Mq] = Mp+q, p, q 2 Z=tZ .
To see an example let n > 0 be an odd integer and let M be
the collection of 2n x 2n real matrices of the form
` '
A S
M = o ,
T -A
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 define a grading as follows:
o 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.
o 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 verified.
The difficulty 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 affect "lower order terms" (e.g.,
terms in Ms-1) which have already been normalized.
We can circumvent the wrap-around problem as follows, assum-
ing V is a (Z=tZ)-cyclically graded vector space (e.g. V := M
as above). We lift V to a Z-graded vector space eV by defining
eVp:= Vp . zp, where the subscript p on Vp is taken mod t, but
that on eVp, and the exponent in zp, is in Z. We think of the
elements in eV as formal Laurent series
A(z) = A-pz-p + . .+.A-1z-1 + A0 + A1z + . . .
where Ap 2 Vp.
38 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
We can now endow V with the structure of a graded R-Lie
algebra by defining the bracket of Azp 2 eVpand Bzq 2 eVqby
[Azp, Bzq] := [A, B]zp+q.
Example (d) above can be viewed as a special case of this con-
struction: regard gl(n, C) as a cyclically graded Lie algebra of
order 1 .
We will study cyclically graded Lie algebras in subsequent pa-
per.
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, ^m2 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+1N;
(c) the condition 0 = [fL(m)] 2 N=F pN implies 0 = [fH (m)] 2
N=F qN for all q p + 1 ;
(d) the condition 0 = [fL(m)] 2 N=F pN implies [f (m)] = [fL(m)] 2
N=F p+1N ; and
(e) Assume p is the smallest integer such that 0 6= [fL(m)] 2 N=F pN
and/or 0 6= [fL(m^)] 2 N=F pN. Then [f (m+m^)] = [fL(m+m^)] =
[fL(m)] + [fL(m^)] 2 N=F p+1N.
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
"first 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 definition.
(c) Since the inclusions F pN F qN for p q induce epimorphisms
N=F pN ! N=F qN 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 definition given in x4, and the normal form algo-
rithm seen in Proposition 4.19, generalize easily to the context of the
initially linear group action ' : G xN ! N under consideration in this
section. Specifically, given s 2 N and ` 2 F 0N define vector spaces
C1s(`) and Vs1+1(`) analogous to (4.12) and (4.13) as follows4:
(5.9) C1s(`) := {g 2 G|f `(g) 2 F s+1N} = {g 2 G|fL`(g) 2 F s+1N}
and
(5.10) Vs1+1(`) = sss+1(f `(C1s(`))) = sss+1(fL`(C1s(`))).
Notice that C1s(`) = C1s(Js(`)) . Indeed with ` = Js(`) + b`we have
g 2 C1s(`) , g . ` = ` mod F s+1(N)
, g . (Js(`) + b`) = Js(`) + g . b`modF s+1(N)
, g . (Js(`)) = Js(`) mod F s+1(N)
, g 2 C1s(Js(`))
As a consequence we see that Vs1+1(`) = Vs1+1(Js(`)) .
Now assume a splitting convention, i.e., that for each s 1 a com-
plement Ys(`) Ns of Vs1(`) has been chosen which depends only on
Js-1(`), hence that
(5.11) Ns = Ys(`) Vs1(`), s 1 .
In particular,
(5.12) Vs1(`) = Ns , Ys(`) = 0.
To involve all non-negative indices in the definition of Vs1(`) define
(5.13) Y0(`) := N0.
Definition 5.14. An element ` = `0 + `1 + . . .2 F 0N 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.
____________
4Recall that for each ` 2 C the mapping f` : G ! N is defined by f` : g !
g . ` - `.
40 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
Proposition 5.15. Suppose ` = `0 + . .+.`s + . . .is in normal form
to order s 0. Write `s+1 = `Ys+1+ `Vs+1 in accordance with the
decomposition (5.11) (with s replaced by s + 1). Choose g 2 C1s(`)
such that sss+1 f `g = -`Vs+1. Then g . ` is in normal form to order
s + 1 and Js(g . `) = Js(`).
The astute reader may have noticed that the negative sign in the
equality sss+1 f `g = -`Vs+1 of the preceding statement does not ap-
pear explicitly in the normal form algorithm described in x4. It does,
however, appear surreptitiously: C1s(`) is defined in terms of ad(`)(g) =
[`, g], and expad (g)(`) has initially linear term [g, `] = -ad (`)(g).
Proof : We have
g . `= ` + g . ` - `
= `0 + . .+.`s + `s+1 + f `(g) + {terms in F s+2N}
= `0 + . .+.`s + `Ys+1+ `Vs+1- `Vs+1+ {terms in F s+2N}
= `0 + . .+.`s + `Ys+1+ {terms in F s+2N},
which by Ys+1(`) = Ys+1(g . `) is in normal form to order s + 1. q.e.d.
Proposition 5.16. Suppose ` = `0 + `1 + . .2.F 0N and `^, ~`2 F 0N
are elements in the G-orbit of ` in normal form to order s 0. Then
Js(^`) = Js(~`).
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 1M
such that
(i) g . ` = ^`.
To verify the case s = 0 write
^`= g . ` = ` + (g . ` - `) = ` + f `(g).
Since f ` preserves filtrations and g 2 F 1G we see that ^`= `0 +
{terms in F 1N}, and this case is established.
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 41
Now assume s 0, that uniqueness holds for s, and write
` = Js(`) + `s+1 + {terms in F s+1N},
^` = Js(^`) + ^`s+1+ {terms in F s+1N}
= Js(`) + ^`s+1+ {terms in F s+1N} .
From (i) and the equality of the s-jets we have g 2 C1s(`), and by
the initial linearity assumption we have
^`= Js(`) + (`s+1 + fL`(g)s+1) + {terms in F s+2N},
hence `s+1 = ^`s+1+ fL`(g)s+1, i.e., `s+1 - ^`s+1= fL`(g)s+1. However, by
definition we have fL`(g)s+1 2 Vs1+1(`), whereas `s+1 - ^`s+12 Ys+1(`)
by the normal form assumption, and `s+1 = ^`s+1follows. q.e.d.
42 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
6. The Spectral Sequence of an Orbit of an Initially
Linear Group Action
Throughout the section R is a field 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 filtration
{F pG}p2Z+ of G as a vector space also provides a filtration 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 Definition 5.6, i.e. for each
` 2 L the mapping
(6.1) f `: G ! L
defined 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
defined by multiplication within G.
With a minor additional hypothesis we can define a covariant functor
from each orbit OC to the category of spectral sequences. The hypoth-
esis 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-1 2 G denote conjugation
by g 2 G. We assume cg is filtration preserving. This is easily seen to
be the case if G is given by the Campbell-Hausdorff formula.
Assumption 6.3. cg(a * b) = cg(a) + cg(b) 2 F pG=F p+1G for all p 2
Z+ and all a, b 2 F pG.
When the group structure is induced by the Campbell-Hausdorff
formula, as in all the examples of the previous sections, the assumption
is an easy consequence of the identity x-1 = -x. Indeed, in this
context each cg is induces the identity mapping on F pG=F p+1G.
Our functor will be a composition. To define the initial factor asso-
ciate to each ` 2 OC the sequence
f`
(6.4) 0 ! G - ! L ! 0
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 43
and to each morphism g 2 OC the commutative diagram
f`
0 ! G -! L ! 0
(6.5) # cg # g . -
fg.`
0 ! G -! 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 {Ep,qr(`)} of the
linear mapping
(6.6) fL`: G ! L.
Now observe, from Assumption 6.3, that the mappings induced by
the morphisms (6.5) are linear in the quotients defining 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 defined directly
from the objects ` 2 OC , whereas the morphisms require the introduc-
tion 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 definitions
(5.9) and (5.10).
Definitions 6.7. For p 1 and r 0 define
(a) Cpr(`) = {g 2 F pG|fL`(g) 2 F p+rL} and
(b) Vrp(`) = ssp+r(fL`(Cpr(`))) Lp+r.
We again have inclusions as seen in (4.26), i.e.,
(6.8)
Cp+r-11(`) Cp+r-22(`) . . .Cpr(`) . . .C1p+r-1(`) F 1G ,
and these in turn induce inclusions
(6.9) V1p+r-1(`) V2p+r-2(`) . . .Vp1+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 finite. The terms appearing in
the spectral sequence {Ep,qr(`)} are easily seen to be related to the
44 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
R-modules appearing in (6.7) as follows:
(68.10) p,-p p p
>> (a) Zr (`) = Cr(`) F G ;
>< p,-p p
(b) Er (`) ssp(Cr(`)) Lp, where ssp : L ! Lp denotes
>> the projection, and
>:
(c) Ep,-p+1r(`) Lp=Vrp-r(`).
We claim that Cpr(`) is a subgroup of G. Indeed, for g 2 G we
have g 2 Cpr(`) if and only if g 2 Lp and g . ` = ` modulo F p+rG. If
a, b 2 Cpr(`) then (a * b) . ` = a . (b . `) = a . ` = ` modulo F p+rG, and
the subgroup assertion follows.
Theorem 6.11. Assuming the standing hypotheses of the section the
following entities are invariants of any fixed G-orbit :
(a) the spectral sequences {Es,tr(`)}r 0 ;
(b) the vector spaces ssp(Cqp(`)) ;
(c) the factor spaces Lp+r=Vrp(`) ;
(d) the vector spaces Vrp(`) ;
(e) the subgroups Cqp(`) .
Moreover, each spectral sequence {Es,tr(`)} is strongly convergent and
for each p 1 we have
(ii) Ep,-p1= ssp({ g 2 F pG | fL`(g) = 0 })
and
(ii) Ep,-p+11= Lp=Vp1-1(`) .
Finally, when the conjugation mappings cg induce the identity map-
pings on each F pG=F p+1G the isomorphisms associated with each of
the invariants in (a)-(e) are given by the identity mapping.
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 finite-dimensional.
Proof :
(a), (b) and (c) : Diagram (6.5) induces an isomorphism of spectral
sequences with inverse induced by the action of g-1 . The isomorphisms
now follow from (6.10).
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 ! Vrp(`) ! F p+rL=F p+r+1L? Lp+r ! Ep+r,-(p+r)+1r(`)? Lp+r=Vrp(`)*
* ! 0
?yg.- ?yg.-
0 ! Vrp(g . `)! F p+rL=F p+r+1L Lp+r ! Ep+r,-(p+r)+1r(g . `) Lp+r=Vrp(g*
* . `)! 0
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.
(e) : It suffices to show that cg : Cpr(`) ! Cpr(g . `) is defined. Howev*
*er,
for a 2 Cpr(`) we have cg(a) . (g . `) = g * a * g-1 . (g . `) = g . (a .*
* `) = g . `
modulo F p+rL, implying cg(a) 2 Cpr(g . `).
For the final convergence statement use (c) and (6.9) and for (i) the
finite dimensionality of G .
q.e.d.
The spectral sequence chart may help clarify the convergence. The
differentials, dr originating in position (p, -p) must eventually be zero
because Ep,-p0is finitely generated. For r large Ep,-p+1ris not in the
image of a differential because the filtration of G is bounded below
by 1 . Notice that the finite-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 computa-
tions and the algorithm in x4 express diagram (3.16) in terms of the
equivalences of (6.10):
f`L p+r
Cpr(`) - ! F L
(6.12) oep,r# # op+r,r
Ep,-pr(`) -dr! Lp+r=Vrp(`)
Note that when the action is expad , ` = `(s) is in normal form to
order s 1 and r = s - p + 1 this becomes
ad(`(s))|Cps-p+1(`(s))
Cps-p+1(`(s)) - ! F s+1L
(6.13) oep,s-p+1# # os+1,s-p+1
ds-p+1 p (s)
Ep,-ps-p+1(`(s)) - ! Ls+1=Vs-p+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
justifies dropping ` from the notation, and we do so when confusion
cannot otherwise result, i.e., we simply write that bottom line as
ds-p+1 p
Ep,-ps-p+1-! Ls+1=Vs-p+1 .
To further ease notation we generally express Lp=Vp1-1as L=V , etc.
Proposition 6.14. For any v 2 F s+1L the following statements are
equivalent:
(a) sss+1(v) 2 Vsp-p+1(`); and
(b) [v] := os+1,s-p+1(v) is killed by the differential ds-p+1.
Distinctions between the two assertions, as well as notational dis-
tinctions between elements of L=V and their representatives in V ,
are often blurred. For example, either of (a) and (b) might be indi-
cated by any one of the following statements: ssp(v) is killed by the
differential; ssp(v) is killed by ad(`); (the class) v is killed by the
differential; 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. Specifically, the
initial terms of the subgroup Cps-p+1(`) G are defined by
(6.15) Ips-p+1(`) := ssp(Cps-p+1(`)) Lp ,
and an element mp 2 Ips-p+1(`) is said to complete in Cps-p+1(`). Com-
paring (6.15) with (6.10c) and assuming the expad -action we see that
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 47
diagram (6.13) can now be written
ad(`(s))|Cps-p+1(`(s))
Cps-p+1(`(s)) - ! F s+1L
(6.16) ssp|Cps-p+1(`(s))
# # os+1,s-p+1
ds-p+1 p (s)
Ips-p+1(`(s)) - ! Ls+1=Vs-p+1(` )
Proposition 6.17. For any element mp 2 Lp the following state-
ments are equivalent:
(a) mp completes in Cps-p+1(`);
(b) ssp(mp) 2 Ips-p+1(`); and
(c) mp survives to Ep,-ps-p+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
Vp1-1(`) can be difficult 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
Ep,-p+11= Lp=Vp1-1and then realize Vp1-1as the kernel of the canonical
linear mapping jp : Lp ! Lp=Vp1-1. This factor space philosophy also
carries over to splitting conventions: a complement Y Lp of Vp1-1
must be the image of a section s : Lp=Vp1-1! Lp of jp, and from this
one sees that to determine Y from Lp=Vp1-1 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 `-(sOj)`, and
this can be accomplished via (6.12) and Theorem 6.11 in terms of the
differentials computed directly from the initially given `. In particular,
the information buried in the differentials is more than sufficient to
calculate the normal form.
In the following examples the actions are derived from the expad -
action, and as a consequence the induced mappings of spectral se-
quences 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.
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 1
0 0 0 0 4 0 6 7
B 0 0 1 0 0 0 0 12C
B C
B 0 0 0 1 0 3 8 0 C
B 0 0 0 0 1 0 0 0 C
` := BB CC ;
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
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 suffices to
compute the actual spectral sequence.)
For purposes of defining 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 := tr(AoB).
From the work in Example 3.18 we know that the only non-trivial
spaces of the E1 -term L=V in filtrations greater than 1 are L2=V11
and L6=V51, hence Lj = Vj1-1 for j = 3, 4, 5 and 7. Without any
additional work we can conclude that the normal form `N = `N0+ `N1+
. . .+ `N7 of ` must have `Nj = 0 for these particular values of j. We
also know from Example 3.18 that each of L2=V11 and L6=V51 has
a single generator, i.e., the images !2 and !6 under j of e26 and
e61 respectively. This information is conveniently summarized by the
diagram
L L L L L
L2 L3 L4 L5 L6 L7
(i) # ss " s # # # # ss " s #
R{!2} 0 0 0 R{!6} 0
wherein the bottom row represents L=V and s is the section of
j : L ! L=V uniquely determined by the condition s(L=V ) = Y .
The first class that needs to be killed is `3 - sss`3 = 3e33, and from
the calculation of the E2-terms in Example 3.18 we see that e1 =
-3(e23+ e22) does the job (as does 3e24, which is the matrix M2 used
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 49
in the Order 3 calculation of Example 4.34). We have
0 1
0 0 0 0 4 0 6 7
B 0 0 1 0 0 0 0 12C
B C
B 0 0 0 1 0 0 8 0 C
B 0 0 0 0 1 0 0 0 C
`(3):= expad (e1)(`) = BB CC ,
B 0 0 0 0 0 1 0 0 C
B C
B 0 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
which is in normal form to order 3, but this is not immediately relevant:
we continue working with the original ` and use the differentials in the
spectral sequence to produce matrices e2 = -4e31 + 8e34, e3 = 12e53
and e4 = -7_2e51 which kill the remaining (` - (s O j)`)-terms. The
normal form
0 1
0 0 0 0 0 0 6 0
B 0 0 1 0 0 0 0 0C
B C
B 0 0 0 1 0 0 0 0C
B C
B 0 0 0 0 1 0 0 0C
B C
B 0 0 0 0 0 1 0 0C
B C
B 0 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= ee4ee3ee2ee1.
Note that the normal form `(3) to order 3 obtained above is not
(quite) the same as the analogous `(3) obtained in Example 4.34, al-
though both have the same 3-jet, thereby illustrating uniqueness up to
order three. The full normal forms do coincide.
Example 6.19. We next illustrate the spectral sequence approach to
normal forms by applying the methods to an example of the type de-
scribed 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 1L where L = T (5),
the upper triangular matrices. N := gl(5, R)=F 1L which may be iden-
tified with TL(5), the lower triangular matrices (with non-zero diagonal
allowed). N is graded by Ni := ss(gl(5, R)i-9), 5 i 9.
The matrix we will analyze is
50 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
0 1
1 0 0 0 0
B 2 1 0 0 0C
B C
` = BB1 2 1 0 0CC.
@ 0 4 2 1 0A
0 6 3 11 1
This is of some interest because it is nongeneric,0i.e., the1lower-left
` ' 1 2 1
0 4
subdeterminants det 0 , det 0 6 and det @ 0 4 2 A all vanish
0 6 3
[GR ]. Our notation follows the previous example with modifications to
adjust for the filtration shift in N .
The basis we use for Gs is given by {esk}, 1 s 5, 1 k 5-s ,
where esk is the 5 x 5 matrix with ak,k+s= 1 and all other entries 0.
The basis for N s is given by {esk}, 5 s 9, 1 k s - 4 , where
esk the 5 x 5 matrix with a9-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
E2 term|is displayed|by|the|following|chart.|| | | |
| | | | | | | | | |
| | | | | | | | | |
_______________________________________________________________||||||||||||*
*||||||||2345678910
| | | | | | | | | |
0 | @ r | @ r | | | | | | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| @| |@ | | | | | | |
-1 | |@ r4 |R@ r | | | | | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| | @| |@ | | | | | |
-2 | | |@ r3 |R@ r | | | | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| | | @| |@ | | | | |
-3 | | | |@ r2 |R@ r | | | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| | | | @| |@ | | | |
-4 | | | | |@ r1R| @ r1|R | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| | | | | @|@ r |@@ r|2R | |
-5 | | | | | | | | | |
_______________________________________________________________||||||||||||*
*||||||||@@
| | | | | | @|@ r |@ @ r|3R |
-6 | | | | | | | | | |
_______________________________________________________________||||||||||@@
| | | | | | | |@ |@ |
| | | | | | | | @ r | @ r|4R
-7 | | | | | | | | | |
_______________________________________________________________||||||||||@@
| | | | | | | | |@ |@
| | | | | | | | | @ r | @ r5R
-8 | | | | | | | | | @ | @
_______________________________________________________________||||||||||
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 51
We compute the differentials as we did in the previous example. The
first non trivial differential is a d6. The calculation of Es,t7is similar to
the previous example and is left to the reader. The only non trivial d6's
are: d6(e12) = 6e73, d6(e14) = -6e72, d6(e22) = 6e84, d6(e23) =
-6e82 and d6(e32) = -6e92 + 6e95. Hence
o E1,-17= R{[e11], [e13]}
o E2,-27= R{[e21]}
o E3,-37= R{[e31]}
o E7,-67= R{[e71]}
o E8,-77= R{[e81], [e83]}
o E9,-87= R{[e91], [e92], [e93], [e94]}
The class [e95] 2 E9,-87was set equal to [e92] by a differential. There
are d7's:
o [e11] 7! [e82], [e13] 7! [-4e82 + 3e84]
o [e21] 7! [-e91 + e93]
The first two differentials defined on filtration 1 are zero in E7 and as a
result we see that [e11] and [e13] survive to E8. In E8 the element [e11]
is represented by e11 + 1_6e23. The precise identification of the represen-
tative of [e11] 2 E8 is necessary for computing d8([e11]). 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.
Specifically, in the language of spectral sequences the calculation of
ad (`(5))(F 5L) + F 6L beginning immediately before (v) in Example
4.34 amounts to calculating d1, and the discussion following the com-
putation of ad(`(5))(m4) is related to computing d2. The fact that one
may choose a matrix in the image of ad(`(5)) which is also in the image
of ad(`(5))(F 5L) allows us to complete a choice of m4 to a matrix in
C42(`(5)), which from the spectral sequence perspective shows that m4
survives to E3. Hopefully this attempt to relate the calculations above
to those in x4 has enlightened rather than confused the reader. A sim-
ilar argument shows that [e13] survives to E8 and is represented by
e13 - 2_3e23 - 1_2e22. d8 may now be determined:
o [e11] 7! [-2e91 + 2e92 - 1_2e93 + 1_2e95 = -5_2e91 + 5_2e9,2]
o [e13] 7! [2e92 - 2e95]
52 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
[e13] survives to E9 and [e91] = [e92] 2 E9.
The resulting Ep,-p+11is summarized by a chart analogous to (i) of
the previous example:
(6.20) L L L L
L5 L6 L7 L8 L9
k k # ss " s # ss " s # ss " s
R{!5} R{!(1)6, !(2)6} R{!7} R{!(1)8, !(2)8} R{!(1)9, !(2)*
*9}
where
o s(!7) = e71
o s(!(1)8) = e81, s(!(2)8) = e83
o s(!19) = 1_4(e9,1+ e92 + e9,3+ e95), s(!29) = e94,
o ss(e72) = ss(e73) = 0
o ss(e8,) = ss(e84) = 0
o ss(e91) = ss(e92) = ss(e93) = ss(e95) = !(1)9
(The splitting is defined as in the previous example.)
We now use the differentials in the spectral sequence to convert ` to
normal form, first noting that ` is already in normal form to order 6.
In degree 7 we have to kill
0 1
0 0 0 0 0
B 0 0 0 0 0 C
B C
B 0 0 0 0 0 C
B C
@ 0 4 0 0 0 A
0 0 3 0 0
(this follows from (6.20)), and by computing the differential d6 one
sees that the matrix
0 1
0 0 0 0 0
B 0 0 1=2 0 0 C
B C
m1 = BB0 0 0 0 0 CC
@ 0 0 0 0 -2=3 A
0 0 0 0 0
will do the job. So the 7th normal form (= m1 . `) is
0 1
1 0 0 0 0
B 5=2 2 0 0 0 C
B C
`(7)= BB 1 2 0 0 0 CC
@ 0 0 0 -19=3 0 A
0 6 0 11 25=3
A SPECTRAL SEQUENCE APPROACH TO NORMAL FORMS 53
We have, leaving the details to the reader:
0 1
1 0 0 0 0
B 5=2 2 0 0 0 C
B C
`(8)= BB 1 0 0 0 0 CC
@ 0 0 0 -19=3 0 A
0 6 0 0 25=3
where `(8)= m2 . `(7),
0 1
0 0 0 0 0
B 0 0 0 11=6 0 C
B C
m2 = BB0 0 0 0 -1=3 CC
@ 0 0 0 0 0 A
0 0 0 0 0
The final step is to find `(9). The term in degree 9 of `(8)has the form
[17_6(e91 + e92 + e93 + e95) - 19_3e94] + (-11_6e91 - 5_6e92 - 17_6e93 + 33_6e9*
*5),
where the term in the parenthesis can be killed by a differential. (The
terms are enclosed in square brackets and parenthesis to distinguish
the components in the splitting. Specifically we have written ` =
[sss(`)] + (` - sss(`)) .) From this point the unique normal form
0 17 1
__ 0 0 0 0
B 5=62 17_ 0 0 0 C
B 6 C
`(9)= BB 1 0 17_6 0 0 CC
@ 0 0 0 -19_3 0 A
0 6 0 0 17_6
is achieved with very little effort; finding the matrix that transforms
`(8)into `(9)requires a bit more work.
First note that
-11_6e91- 5_6e92- 17_6e93+ 33_6e95= -28_6(e91- e92) - 17_6(e93- e91) + 11_3(e95*
*- e92) ,
and that the terms in the parenthesis in the right side of the equality
are hit by differentials, e.g., -28_15(e11 + 1_6e23) 7! -28_6(e91 - e92). In th*
*is
way we determine that the matrix
0 1
0 28_15-17_60 0
B 0 0 0 0 11_C
B 12C
m3 = BB0 0 0 0 14_45CC
@ 0 0 0 0 0A
0 0 0 0 0
satisfies m3 . `(8)= `(9).
54 MARTIN BENDERSKY AND RICHARD C. CHURCHILL
We now illustrate Theorem 6.11. If we compute the differentials
in the spectral sequence E*,*r(`(9)) we find d7(e13) = 0, d8(e13) = 0
and d8(e11) = -5_2e91 + 5_2e92. In the quotients that define E*,*rthese
differentials are identical to the corresponding differentials in E*,*r(`).
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
B b21 a 0 0 0C
B C
N = BBb31 0 a 0 0CC
@ 0 0 b43 b 0A
0 6 0 0 a
is the normal form. Now compute the spectral sequence E*,*r(N). The
invariance of this sequence, in particular that of the differentials, com-
pletely 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
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E-mail address: mbenders@math.hunter.cuny.edu
E-mail address: rchurchi@math.hunter.cuny.edu
Hunter College and Graduate Center, CUNY, New York, NY 10021