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 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 fields and Hamiltonians, J. Diff. Eqns., 78 (1989) 33-52. [BC] A. Baider, R.C. Churchill, The Campbell-Hausdorff group and a polar de- composition of graded algebra automorphisms, Pacific J. Math, 131, (1988), 219-235. [C] K.-T. Chen, Equivalence and decomposition of vector fields about an elemen- tary critical point, Am. J. Math. 85 (1963), 693-772. [CK] R.C. Churchill and M. Kummer, A Unified 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. Differential Equations, 49 (1983), 359- 414. [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 Alg'ebrique et Th'eorie 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. Diff. 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. Diff. Eqns., 192, (2003), 536-552. [Sa2]J. Sanders, Normal form in filtered 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: mbenders@math.hunter.cuny.edu E-mail address: rchurchi@math.hunter.cuny.edu Hunter College and Graduate Center, CUNY, New York, NY 10021