Ann. Global Anal. Geom. Vol. 7, No. 3 (1989), 163-169
A Cohomology for Vector Valued Differential Forms PETER W. MICHOR
and HUBERT SCHICKETANZ
Abstract. A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr6licher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of "traceless" vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed. 1. Notation 1.1 The Frolicher-Nijenhuis bracket. Let M be a smooth manifold of dimension m throughout the paper. We consider the space Q(M; TM) =
Q2k(M; TM) of all tangent k=O
bundle valued differential forms on M. Below K and L will be elements of Q(M; TM) of degree k and , respectively. It is well known that Q2(M; TM) is a graded Lie algebra with the so called Frolicher-Nijenhuisbracket [,]: f?(M; TM) x Q'(M; TM) -* Q+'(M; TM). For its definition, properties, and notation we refer to [Mi, 1987]. 1.2. In the investigation of the Lie algebra cohomology of the graded Lie algebra (Q(M; TM), [,]) in [Sch, 1988] the following exterior graded derivation of degree 1 appeared: 6: 2k'(M; TM) -
Q+t(M; TM).
Before its definition we need another operator. Let the contractionor trace c: 2Qk(M; TM) -k -'(M) be given by c(qp ® X) = ixT, linearly extended. We also put E JQk(M; TM):=
c for clear reasons which become i-kn lemma 2.1 below.
for reasons which become clear in lemma 2.1 below. 12 Annals Bd. 7, Heft 3 (1989)
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P. W. Michor and H. Schicketanz
Then let 6(K):= (- 1)k - dc(K) A I, where I = IdrMEc'(M; TM) is the generator of the center of the Fr6licher-Nijenhuis algebra. This operator is a derivation with respect to the Fr6licher-Nijenhuis-bracket, so we have 6([K, L]) = [6(K), L] + (-1)k [K, 6(L)]; furthermore 6 o 6 = 0. These properties are proved in [Sch, 1988] and are straightforward to check. 2. The cohomology space H(Q(M; TM), 6) 2.1. Lemma. The mappingj := ( A I): Qk(M) the following diagram commutes for k < m:
f' (a)
ff(M; TM)
(M)
&2k(M)
J
'(M; TM) is a right inversefor and
-Qk-(M) A
m1 i6
dl
+
dl
g2k+l(M;
TM)
e
ik(M)
Proof. Both operators are local and in a coordinate system we write I = idM = dx i ( a,. Let E 2k(M), then we have c(q A I) = c (i ( A dx'i
ai)
i (i,,(p A dx i + (-l)k p A i,, dx')
=
= c((- 1) k - 1 kp + (- 1) m(p) = (--1) (m - k) (q. The rest is a consequence of this. 2.2. Since c oj = Id, the mapping P := j o : Q(M; TM) - Q(M; TM) is a projection, so Po =P and (a)
Q(M; TM) = im P K = (K)
A
ker P,
I + K'.
Note that j is injective; this has the following consequences: K' in the decomposition (a) is characterized by (K') = 0. Moreover 6(K) = 0 if and only if dc(K) = 0. Finally ker P = ker c = ker c; we will denote this space by m
(b)
W(M) = (
k(M).
k=O
Then Wk(M) = {K E Q2k(M; TM): c(K) = 0} = C-(Ek(M)) is the space of smooth sections of a certain natural vector bundle over M, namely the subbundle ker c of AkT*M ® TM. If(U, x) is a chart on M, the sections dxi" A ... A dxik ® aj with i < ... < i and j il for all I give a local framing of this bundle. Note that '°0(M) = 3E(M), the space of all vector fields, and that W'(M) is the space of all traceless endomorphisms TM - TM. For this reason elements of t(M) will be called traceless vector valued differential forms.
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165
Note that I m(M) = 0 since : Qm(M; TM) - 0"-'(M) is a linear isomorphism. 23. Let us define the natural bilinear concomitant S: 2k(M; TM) x Q'(M; TM) _ k+ '- 2(M) by S(p ® X, V)® Y) = iy( A ixhw for decomposable vector valued forms. Lemma. Then we have c([K, LI) = (- 1) O(K) c(L)-
(- 1)
l
(L) c(K) - (- l)k dS(K, L).
Proof. Since both sides are local in K and L we may assume that K = L = p ® Y are decomposable. Then by formula [Mi, 1987, 1.7.7] we have (a)
[K, L] = tp A p®[X, Y] + (p A O(X) tp + (-l)k(dq
A ixtpW
Y + i
Y-
(Y)
(a)
X and
A ^p ( X
A dp ® X).
This implies the lemma by a straightforward computation. 2.4. Theorem. ((M),
qp
X
[, ]) is a graded Lie algebra, where
[K, L := [K, L] -
k m-k-l+l
1
dS(K, L) AI.
It is a quotient of a subalgebra of (Q2(M; TM), [,]). The bracket [,]c is a natural bilinear differential concomitant of order 1, that means f*[K, L]C = [f*K, f*L] for each local diffeomorphism f This theorem will be proved jointly with theorem 2.5 below. 2.5. Theorem. The cohomology ofthe gradedLiealgebraQ2(M; TM) is decomposed into: (a)
Hk(f2(M; TM), )
H[d 1'(M) D (ker c k = HR 1(M)
Wk(M)
The inducedbracket [,]: Hk(Q2(M; TM), ) x H'(Q(M; TM), ) -* Hk+'(92(M; TM), ) corresponds to the direct product of the graded Lie algebra ((M), [,]C) with the abelian algebra (H* '(M), 0). Proof of 2.4 and 2.5. By diagram 2.1.a we have induced mappings in cohomology j* = ( A)*: H -'(M) - H*(fQ(M; TM), 6) and *: H*(M; TM), 6) - H-(M); again c# oj* = id, so P = j o 6* is also a projection in cohomology, and we have the decomposition Hk(Q2(M; TM), 6)
(im p#)k
e
(kerP#)k = H'
1(M)'
i
k(M).
Here we used that j*: HdR'(M) -(im P*)k is a linear isomorphism, and that kerP* = kerP since = +dc( ) A I implies im3 c imj = imP and kerP = ker 6 c ker 6. Since the differential is a graded derivation, ker is a graded Lie subalgebra of (Q2(M; TM), [,]), the space im is an ideal in ker and the cohomology space H(Q(M; TM), 6) is a graded Lie algebra. 12'
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P. W. Michor and H. Schicketanz
It remains to investigate the induced bracket. Let K = c(K) A I + K' E (ker 6)k c Q(M; TM) = (im P)k e (ker P)k and similarly L = (L) A I + L', then by using formula [Mi, 1987, 1.7.6] we may compute as follows: [K, L] = [(K) A I, E(L) =
A
I] + [(K)
A
I, L'] + [K', E(L)
A
] + [K', L']
- (- l)kl O(e(L) A ) (K) A I + (- 1)k dc(K) A i(E(L) A I) l l)k
+ 0 - (-
(L')E(K) A I + (- 1) d(K) A iL'
+ O + O(K')e(L) A I- (-1)kl+d"'(L)A iK' + [K', L' = O(K') E(L)
A
I - (-
1)k
(L') (K) A I + [K', L'],
since by the definition of we have d(K) = 0 and d(L) = 0 for k, I < m. By 2.2 we have e(K') = 0 and E(L') = O,so by 2.2, 2.3, and 2.4 we have (b)
[K', L'] = E([K', L']) (-1)'
m-k-I
A
I + [K', L']'
+ 1 dS(K', L')
A
I + [K', L]' .
Since (K) is closed, (L') E(K) = i(L') de(K) - (- 1) - di(L') c(K) is exact, likewise e(K') (L) is exact. Therefore in H(Q(M; TM), ) = H*-'(M) E (M) we have [a + K', fi + L'] = [K', L'i. So H*- i(M) is an abelian ideal of H*(Q(M; TM), 6), the subspace ((M), [,]c) is an ideal of H*(Q(M; TM), 6), and a quotient of the graded Lie (M). The differential concomitant [,]' is natural since its algebra j(Z*-l(M))( composants (in 2.4) are all natural. ·
e
3. Extensions of the graded Lie algebra W(M) 3.1. Graded Lie subalgebrasof Q(M; TM). Let m-1
m-1
*- (M) =
3
k=O
k(M),
mn-I
B*- (M) = c( Bk(M),
Z*- (M) =
k=O
Zk(M),
m-1
H*-'(M) = (
*kk=O
H'dR(M)
=O
be the graded spaces of de Rham forms, cycles, boundaries, and cohomology classes, respectively, where we exchanged the top degree spaces for 0. Theorem. With this notation we have: (1) B*-'(M) A I is an abelian ideal in Q*(M; TM). (2) Z*- i(M) A I is an abelian ideal in Q*(M; TM).
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167
(3) fQ*- (M) A I is a graded Lie subalgebra of Q*(M; TM). Therefore (*- (M), [,]) is a gradedLie algebra, where the bracket inducedfrom the Friilicher-Nijenhuisone looks asfollows for p E 2(M) and p E 2'(M):
[qp, p] = ()k- 1 (d A =()1 (e ())
-
- (A i
1
-
)(k (
1)(1- ) dp A kp) )(k )(l- 1) () W
A
i).
Proof: Straightforward computations using [Mi, 1987, 1.7]. 3.2. Extension of se(M). If we collect all relevant parts in the proof of theorem 2.5, we get for K = (K) A I + K' E(Z k - l(M) A I)® k(M) and L = E(L) A I + L' E (Z - 1(M) A 1) 33 V(M) the following formula: (a) where a:
[K, L] = ((K') k(M)xW(M)
E(L)-
(- 1)k l (L') E(K) + a(K', L')) A I + [K', L']',
Bk + - (M) c Zk+l - (M) is defined by
a(K', L') =
dS(K', L')
Now by 3.1.2 we have an exact sequence of graded Lie algebras (b)
0 -- Z*-(M)
(Z*-(M)%A 1)
C(M) -
(M)
0,
which describes an abelian extension of W(M). By 2.4 we have (c)
O([K', L']c) =
([K', L'] - a(K', L') A I) = [(K'), O(L')]- a(K', L') A O(I) I Z(M) = [(K'), O(L')].
So O gives a graded Lie module structure over (M) to Z*- (M) and consequently a is a cocycle for the graded Lie algebra cohomology of ((M), [,]C) with coefficients in the W(M)-module Z*- '(M), because the second cohomology classifies equivalence classes of abelian extensions, just as in the non-graded case. For the convenience of the reader we sketch this in 3.3 below. A direct check shows that indeed the cocycle equation for a is valid and is equivalent to the graded Jacobi identity for ((M), [, ]'). 3.3. Cohomology of gradedLie algebras. In the following we write down the definitions for the graded cohomology of a Z-graded Lie algebra L = ( Lk with coefficients in a keZ
V k is a graded vector space and
graded L-module V. So V =
0: L - End (V) is a
keZ
homomorphism of graded algebras (of degree 0, where the bracket on End (V) is the graded commutator). Our definitions are different from but equivalent to those of [Le], which are also used in [Sch]; we obey Quillen's rule strictly. Let AP(L; V)9 be the space of all p-linear mappings : L x ... x L - V, which are of degree q, i.e., (X 1 , ... , XI) E V + +x, + for Xl E L, and which are alternating in the sense that reacts to interchanging Xi and Xi+x with the sign -(-1) xi' ' *'.
168
P. W.Michor and H. Schicketanz
Now let us define the differential a: AP(L; V) -* AP+ 1(L; V)" by
(a) (Xo, ... , xp) = Y, (-1),+x' + E(-l)j
+ -
(Xi) (Xo ....X, .... Xp) l
x
ix ' ([Xj, Xj], Xo, .... Xi .... Xj ... Xp)
i
where a, = i + x,(xo + ... + x,_ ), and where the brace over a symbol means that it
has to be deleted. Then one may check that a a = 0, and we denote by HP(L; V)" ·= HP(A*(L; V)", a) the resulting cohomology and call it the graded cohomology of the graded Lie algebra L Theorem. H2 (L; V)O is isomorphic to the set of equivalence classes of abelian extensions of L by V.
The proof of this is completely analogous to the non-graded case after the insertion of some obvious signs. 3.4. The Nijenhuis-Richardson bracket. Recall from [Ni-Ri, 1967] or [Mi, 1987], that there is a natural graded Lie algebra structure on 2*- '(M; TM), given by [K, L]A = i(K) L-
- -(k (- 1)
1) (
1)
i(L) K .
Theorem. For the Nienhuis-Richardson bracket we have (1) C*-1 (M) is a graded Lie subalgebra of (Q2*-(M; TM), [,]^). 1 (2) 2*(M) A I is a graded Lie subalgebraof (2*-(M; TM), [,]^). Therefore (2*(M), [, ] ) is a gradedLie algebra, where the induced bracket looks asfollows for qp E 2k(M) and p E (M): [qp, ]
= (I - k) =
AW
A A iWp - (-1)klW
A i
·
None of these two subalgebras is an ideal, so there is no extension. For the structure of the whole algebra in terms of the subalgebras see [Mi, 1988]. Compare the formulas here and in 3.1. They rise the question whether there is a general procedure behind them. References [Fr-Ni] A. FROLICHER, A. NuENHUIS, Theory of vector valued differential forms. Part I., Indagationes Math 18 (1956), 338-359. [Ko-Mi] I. KOLAA, P. W. MICHOR, Determination of all natural bilinear operators of the type of the Fr6licher-Nijenhuis bracket, Proceedings of the Winter School on Geometry and Physics, Srni 1987, Suppl. Rendiconti Circolo Mat. Palermo, Serie II, 16(1987), 101-108. [Le] P. B. A. LECOMTE, Applications of the cohomology of graded Lie algebras to formal deformations of Lie algebras, Letters in Math. Physics 13 (1987), 157-166. [Mi] P. W. MICHoR, Remarks on the Fr61licher-Nijenhuis bracket, in: Proceedings of the Conference on Differential Geometry and its Applications, Brno 1986, D. Reidel, 1987.
[Mi]
P. W. MICHOR, Knit products of graded Lie algebras and groups, preprint 1988.
A cohomology for vector valued differential forms [Ni-Ri] [Sch]
169
A. NJENHUIS, R. RICHARDSON, Deformation of Lie algebra structures, J. Math. Mech. 17 (1967), 89-105. H. ScHIcKETANz, On derivations and cohomology of the Lie algebra of vector valued forms related to a smooth manifold, Bul. Soc. Roy. Sc. de Liege, 57e annie, 6 (1988), 599-617.
Peter W. Michor Institut fiir Mathematik Universitit Wien Strudlhofgasse 4 A-1090 Wien Hubert Schicketanz University de Liege Institut de Mathematique Avenue des Tilleuls, 15 B-4000 Liege
(Received October 11, 1988)