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On the cohomology of vector ?elds on parallelizable manifolds

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arXiv:0705.3382v1 [math.RT] 23 May 2007

On the cohomology of vector ?elds on parallelizable manifolds

Yuly Billig, Karl-Hermann Neeb

Abstract. In the present paper we determine for each parallelizable smooth compact manifold p M the cohomology spaces H 2 (VM ,?M ) of the Lie algebra VM of smooth vector ?elds on M with p values in the module ?M =?p /d?p?1 . The case of p=1 is of particular interest since the gauge M M 1 algebra C ∞ (M,k) has the universal central extension with center ?M , generalizing a?ne Kac-Moody 1 2 algebras. The second cohomology H (VM ,?M ) classi?es twists of the semidirect product of VM with 1 the universal central extension C ∞ (M,k)⊕?M . Keywords: Lie algebra of vector ?elds, Lie algebra cohomology, Gelfand-Fuks cohomology, extended a?ne Lie algebra MSC 2000: 17B56, 17B65, 17B68

One of the most important insights in the theory of a?ne Kac–Moody Lie algebras is that they can all be realized as twisted or untwisted loop algebras. In the untwisted case, this realization starts with the Lie algebra of maps from a circle S1 to a ?nite-dimensional complex simple Lie algebra k, L(k) := C[t, t?1 ] ? k, then proceeds with the universal central extension C ?→ L(k) → L(k), →

d and completes the picture by adding a derivation d acting by t dt on L(k) and, accordingly, on the central extension. In the representation theory of a?ne algebras, an important role is played by the Virasoro algebra, which emerges via the Sugawara construction. At the Lie algebra level, the Witt algebra of vector ?elds on a circle, d := Der(C[t, t?1 ]) acts on L(k), so that we may form the semidirect product g := L(k) ? d. To get the Virasoro algebra, we need to twist the above semidirect product by a 2 -cocycle τ ∈ Z 2 (d, C), which leads to the a?ne-Virasoro Lie algebra gτ with the bracket

[(x, d), (x′ , d′ )] = ([x, x′ ] + d.x′ ? d′ .x + τ (d, d′ ), [d, d′ ]), which contains both a?ne Kac-Moody algebra and the Virasoro algebra as subalgebras. The theory of a?ne Kac–Moody algebras has an analytic side, where one replaces the algebra C[t, t?1 ] of Laurent polynomials by the Fr?chet algebra C ∞ (S1 , C) of complex valued e functions on the circle. In this context one also obtains a one-dimensional central extension and the role of the Witt algebra is played by the Fr?chet–Lie algebra VS1 of smooth vector ?elds on e the circle, which also has a one-dimensional central extension. For an exposition of these ideas we refer to the monograph of Pressley and Segal [PS86]. Let us discuss the generalization of this construction from S1 to the case of an arbitrary ∞ C -manifold M . The gauge Lie algebra of k-valued functions on M , LM (k) = C ∞ (M, k), endowed with the pointwise de?ned Lie bracket and the natural Fr?chet topology, has the e universal central extension with the central space ? (M, C) := ?1 (M, C)/d?0 (M, C)

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2

Yuly Billig, Karl-Hermann Neeb

(cf. [Ka84], [Ma02]). The Lie bracket on the Lie algebra LM (k) = C ∞ (M, k) ⊕ ? (M, C) is given by the formula [f1 ? g1 , f2 ? g2 ] = f1 f2 ? [g1 , g2 ] + (g1 |g2 )[f2 df1 ], where f1 , f2 ∈ C ∞ (M, C), g1 , g2 ∈ k, (·|·) is the Cartan–Killing form on k and [α] denotes the 1 class of a 1 -form α in ? (M, C). The Lie algebra VM of smooth vector ?elds on M acts on both the gauge algebra C ∞ (M, k) 1 and on ? (M, C) by the Lie derivative. This action is compatible with the above central cocycle, allowing us to consider the semidirect product of VM with LM (k): g = C ∞ (M, k) ⊕ ? (M, C) ? VM . However the interplay between a?ne Lie algebras and the Virasoro algebra suggests that it will 1 2 be natural here to twist the Lie bracket in g by means of a 2-cocycle τ ∈ Zc (VM , ? (M, C)), resulting in the family of Lie algebras gτ = C ∞ (M, k) ⊕ ? (M, C) ⊕τ VM , so that the equivalence classes of twistings are classi?ed by the cohomology space

1 1 2 Hc (VM , ? (M, C)) ? Hc (VM , ? (M, R)) ?R C. = 2 1 1 1

In the present paper we calculate this space for parallelizable manifolds M , relying heavily on results on the cohomology of Lie algebras of vector ?elds with values in di?erential forms by Gelfand–Fuks, Hae?iger and Tsujishita. We calculate this space using the short exact sequences

1 2 0 → HdR (M, R) ?→ ? (M, R)) → BdR (M, R) → 0 1

and

1 0 → BdR (M, R) ?→ ?1 (M, R) → ? (M, R) → 0,

1

and then applying the corresponding long exact sequence in cohomology. Hence we need detailed knowledge on the cohomology of VM with values in di?erential forms. In the following M denotes an N -dimensional parallelizable compact manifold. The main goal of this paper is to describe for each p ∈ N the second cohomology spaces

2 Hc (VM , ? (M, R)) p

for p ∈ N,

? (M, R) := ?p (M, R)/d?p?1 (M, R)

p

with respect to the natural action of the Lie algebra VM of smooth vector ?elds. For reasons described above, our main interest lies in the case p = 1 . For N = 1 and connected M we have 1 2 M ? S1 and ? (M, R) ? R is a trivial module. Then Hc (VS1 , R) ? R describes the central = = = extensions of VS1 , and a generator corresponds to the Virasoro algebra. For N ≥ 2 , we ?nd that

1 2 Hc (VM , ? (M, R)) ? HdR (M, R) ⊕ R2 . = 3

Here we use that each closed 3 -form ω yields an ? (M, R)-valued 2 -cocycle by ω [2] (X, Y ) := [iY iX ω] (cf. [Ne06a]). In terms of a trivializing 1 -form κ ∈ ?1 (M, RN ), the other two cocycles can be described as follows. De?ne θ: VM → C ∞ (M, glN (R)) by LX κ = ?θ(X) · κ, Ψ1 (X) := Tr(θ(X)) ∈ ?0 (M, R) = C ∞ (M, R), Ψ1 (X) := Tr(dθ(X)) ∈ ?1 (M, R),

1

On the cohomology of vector ?elds on parallelizable manifolds

3

and

Ψ2 (X, Y ) := [Tr(dθ(X) ∧ θ(Y ) ? dθ(Y ) ∧ θ(X))] ∈ ? (M, R).

2 Then Ψ1 ∧ Ψ1 and Ψ2 provide the two additional generators of Hc (VM , ? (M, R)). In the case when M is a torus TN , we give explicit formulas for these cocycles in coordinates. For the toroidal Lie algebras, the cocycles Ψ2 and Ψ1 ∧ Ψ1 were discovered in the representation theory of these algebras ([EM94], [L99], see also [BB99], [Bi06]). It turns out here that it is easier to construct representations of gτ for a non-trivial cocycle τ , rather than for the semidirect product g. For the toroidal Lie algebras it would be natural to work in the algebraic setting, considering the algebra A of Laurent polynomials in N variables as the algebra of functions on TN (Fourier polynomials), instead of C ∞ (TN , C). Although the cocycles that we get here are well-de?ned in this algebraic setting, our results do not solve the problem of describing the algebraic cohomology H 2 (Der(A), ?1 (A)/dA). The reason for this is that our proof is based on the results of Tsujishita and Hae?iger, while the algebraic counterparts of these have not been established. Toroidal Lie algebras are also closely related to the class of extended a?ne Lie algebras ([Neh04], [BN06]). According to a result of [ABFP05], most of the extended a?ne Lie algebras are twisted versions of toroidal Lie algebras. However, instead of the Lie algebra of vector ?elds VTN , in the theory of extended a?ne Lie algebras one must use the Lie algebra of divergence div zero vector ?elds VTN (or a subalgebra). Thus it would be also interesting to describe the div space H 2 (VTN , ? (TN , R)). This question remains open. We point out here that the restriction div of the cocycle Ψ1 ∧ Ψ1 to VTN vanishes, while Ψ2 does not. Also we note that the cocycles 3 corresponding to HdR (M, R) are not relevant for the theory of extended a?ne Lie algebras either, because of the additional restriction that the cocycle should vanish on the subalgebra of degree zero derivations, so that this subalgebra remains abelian. p The cohomology of the subcomplex Cloc (VM , ?p ) of local cochains, i.e., cochains f ∈ p p Cc (VM , ?M ) that are di?erential operators in each argument is easier accessible than the full complex. In view of [Tsu81], the di?erence fully comes from the case p = 0 . In this case, the ? description of Hloc (VM , FM ) is facilitated signi?cantly by the results of de Wilde and Lecomte ? ([dWL83]) who show that the cohomology of the di?erential graded algebra Cloc (VM , FM ) ? coincides with the cohomology of the subalgebra generated by the image of ?M , the image of the Chern–Weil homomorphism 2? χ1 : Sym? (glN (R), R)glN (R) → Cloc (V(M ), FM ) 1 1

1

corresponding to a connection on the frame bundle J 1 (M ) of M (which yields cocycles depending on ?rst order derivatives), and the image of an algebra homomorphism

? χ2 : C ? (glN (R), R) → Cloc (V(M ), FM )

whose image consists of cocycles if the connection is ?at. If all Prontrjagin classes of M vanish, then this results in an isomorphism

? Hloc (VM , FM ) ? HdR (M, R) ? H ? (glN (R), R) ? HdR (M, R) ? C ? (glN (R), R)glN (R) . = ? = ? 2 As our results show, for N > 1 , all classes in Hc (VM , ? (M, R)) can be represented by local 2 2 cochains, but the generator of Hc (VS1 , R) ? Hc (VS1 , C ∞ (M, R)) is non-local because it involves an integration. Some of Tsujishita’s results have been generalized by Rosenfeld in [Ro71] to other classes of irreducible primitive Lie subalgebras of VM . It would be of some interest to see if these results 1 could be used to obtain the cohomology of these Lie algebras with values in ?M . p

4

Yuly Billig, Karl-Hermann Neeb

Notation and conventions If V is a module of the Lie algebra g, we write C p (g, V ) for the corresponding space p-cochains, dg : C p (g, V ) → C p+1 (g, V ) for the Chevalley–Eilenberg di?erential, B p (g, V ) = dg (C p?1 (g, V )) for the space of p-coboundaries, Z p (g, V ) = ker(dg : C p (g, V ) → C p+1 (g, V )) for the space of p-cocycles and H p (g, V ) for the Lie algebra cohomology. If g is a topological p Lie algebra and V a continuous g-module, then Cc (g, V ) etc. stands for the space of continuous cochains. For x ∈ g we have the insertion operator ix : C p+1 (g, V ) → C p (g, V ) and there is a natural action of g on C p (g, V ), which is denoted by the operators Lx , satisfying the Cartan relation Lx = dg ? ix + ix ? dg . For a subalgebra h ≤ g we write C p (g, h, V ) := {α ∈ C p (g, V ): (?x ∈ h) ix α = 0, ix(dg α) = 0} for the relative cochains modulo h, and accordingly B p (g, h, V ), Z p (g, h, V ) and H p (g, h, V ). If G is a group, we denote the identity element by 1, and for g ∈ G, we write λg : G → G, x → gx for the left multiplication by g , ρg : G → G, x → xg for the right multiplication by g , mG : G × G → G, (x, y) → xy for the multiplication map, and ηG : G → G, x → x?1 for the inversion. If M is a smooth manifold, we write VM for the Lie algebra of smooth vector ?elds on M , FM := C ∞ (M, R) for the algebra of smooth real-valued functions on M , ?k := ?k (M, R), the M space of smooth real-valued p-forms on M , and

k ZM := ker(d |?k ),

M

k?1 k BM := d?M ,

k k HM := HdR (M, R),

and

k ?M := ?k /BM . M

k

We write T (p,q) (M ) := T (M )?p ? T ? (M )?q for the tensor bundles over M and Γ(T (p,q) (M )) for their spaces of smooth sections, i.e., the (p, q)-tensor ?elds.

I. Crossed homomorphisms of Lie algebras and pull-backs

In this short ?rst section, we collect some basic facts on crossed homomorphisms of Lie algebras which provide the tools used throughout the forthcoming sections. The main point is Theorem I.7 on maps in Lie algebra cohomology de?ned by crossed homomorphisms. Let h and n be two Lie algebras, with h acting on n by derivations, i.e., we are given a Lie algebra homomorphism τ : h → Der(n). Example I.1. (a) Our primary example would be h = VM and n = C ∞ (M, g), where g is a ?nite-dimensional Lie algebra and the bracket on n is de?ned pointwise. Then (τ (v).f )(m) := (v.f )(m) := df (m)v(m) de?nes a Lie algebra homomorphism from VM to Der(C ∞ (M, g)). (b) If A is a commutative associative algebra and g a ?nite-dimensional Lie algebra, then n := A ? g carries a Lie algebra structure de?ned by [a ? x, a′ ? x′ ] := aa′ ? [x, x′ ]. Then we obtain an action of h := Der(A) on g by τ (D).(a ? x) := (D.a) ? x.

On the cohomology of vector ?elds on parallelizable manifolds

5

De?nition I.2.

A linear map θ : h → n is called a crossed homomorphism if θ([x, y]) = [θ(x), θ(y)] + τ (x)θ(y) ? τ (y)θ(x) for all x, y ∈ h.

Remark I.3. (a) Note that θ is a crossed homomorphism if and only if the map (θ, idh ): h → n ? h is a homomorphism of Lie algebras. In terms of the Lie algebra cohomology of h with values in the h-module n, the condition that θ is a crossed homomorphism can be written as (dh θ)(x, y) + [θ(x), θ(y)] = 0, i.e., θ satis?es the Maurer–Cartan equation dh θ + 1 [θ, θ] = 0. 2 (b) If h ? n ?η s is a semidirect product and the action of h on n is de?ned by τ (n, s) := = η(s), then [(n, s), (n′ , s′ )] = ([n, n′ ] + η(s).n′ ? η(s′ ).n, [s, s′ ]) implies that θ: h → n, (n, s) → n is a crossed homomorphism. (c) The kernel of any crossed homomorphism is a subalgebra. Remark I.4. Let ρ: h → n be a homomorphism of Lie algebras. Then we obtain on n an action of h by τ (x).y := [ρ(x), y]. Now a linear map θ: g → n is a crossed homomorphism if and only if ρ′ := ρ + θ is a homomorphism of Lie algebras. This follows directly by comparing ρ′ ([x, y]) = ρ([x, y]) + θ([x, y]), with [ρ′ (x), ρ′ (y)] = [ρ(x), ρ(y)] + [θ(x), ρ(y)] + [ρ(x), θ(y)] + [θ(x), θ(y)]. It follows in particular that θ := ?ρ is a crossed homomorphism. Let us show how a crossed homomorphism may be used to pull back cocycles on n to cocycles on h. First, we need to de?ne the notion of an equivariant cochain. De?nition I.5. Let V be a module for the Lie algebra h and a trivial module for n. A cochain ? ∈ C k (n, V ) is called equivariant if

k

x?(y1 , . . . , yk ) =

j=1

?(y1 , . . . , τ (x)yj , . . . , yk )

for all

x ∈ h, y1 , . . . , yk ∈ n,

which is equivalent to Lx ? = 0 for each x ∈ h. Since [Lx , dn ] = 0 holds for each x ∈ h on C ? (n, V ), we have:

? Lemma I.6. Equivariant cochains form a subcomplex Ceq (n, V ) in C ? (n, V ).

By de?nition, the subcomplex of equivariant cochains yields the equivariant cohomology ? Heq (n, V ). Identifying n with a subalgebra of the semidirect product n ? h, we may identify C p (n, V ) with the subspace {α ∈ C p (n ? h, V ): (?x ∈ h) ix α = 0}. Hence the relative cochain space C p (h ? n, h, V ) := {α ∈ C p (n ? h, V ): (?x ∈ h) ix α = 0, ix (dα) = 0}

p can be identi?ed with Ceq (n, V ) because if ix α = 0 holds for each x ∈ h, then Lx α = ix dα , so that invariance is equivalent to α ∈ C p (n ? h, h, V ) (cf. [Fu86, p. 16]).

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Yuly Billig, Karl-Hermann Neeb

Theorem I.7. Let V be an h-module and consider it as a trivial n-module. Further let θ be a crossed homomorphism of Lie algebras θ : h → n. Then the map

? θ? : Ceq (n, V ) → C ? (h, V ),

? → ? ? (θ × . . . × θ)

is a morphism of cochain complexes, hence induces a linear map of cohomology spaces ? θ? : Heq (n, V ) → H ? (h, V ).

? Proof. We have seen above that we may identify the complex (Ceq (n, V ), dn ) with the relative Lie algebra subcomplex

(C ? (n ? h, h, V ), dn?h ) ? (C ? (n ? h), dn?h ).

p For ? ∈ Ceq (n, V ), we write ? for the corresponding element of C p (n ? h, V ). Then we have

θ? ? = (θ, idh )? ?, and the assertion follows from the fact that (θ, idh ): h → n ? h is a morphism of Lie algebras, hence induces a morphism of cochain complexes C ? (n ? h, V ) → C ? (h, V ).

Some applications to the Lie algebra of vector ?elds In this subsection M denotes a ?nite-dimensional smooth manifold. Proposition I.8. homomorphism A g-valued di?erential form θ ∈ ?1 (M, g) de?nes an FM -linear crossed VM → C ∞ (M, g), X → iX θ

1 if and only if θ satis?es the Maurer–Cartan equation dθ + 2 [θ, θ] = 0.

Proof. Since the exterior di?erential on ?? (M, g) coincides with the VM -Lie algebra di?erential, this is an immediate consequence of Remark I.3(a). Recall that the left Maurer–Cartan form κG ∈ ?1 (G, g) of a Lie group G with Lie algebra g is de?ned by κG (v) := T (λ?1 )v for v ∈ Tg (G). g Corollary I.9. (cf. [BR73], Th. 3.1) If M is a smooth manifold, g a Lie algebra and κ ∈ ?1 (M, g) satis?es the Maurer–Cartan equation, then we have a map

? ζ: C ? (g, R) → Cc (VM , FM ),

ζ(ω)(X1 , . . . , Xp ) := ω(κ(X1 ), . . . , κ(Xp )),

whose range consists of FM -multilinear cocycles represented by di?erential forms. In particular, we obtain an algebra homomorphism

? H ? (g, R) → Hc (VM , FM ).

Proof. In view of Theorem I.7, it only remains to verify that ζ is compatible with the multiplication of cochains, but this is an immediate consequence of the de?nition. Corollary I.10. If G is a Lie group with Lie algebra g and κG ∈ ?1 (G, g) the left Maurer– Cartan form, then θ: VG → C ∞ (G, g), X → iX κG is a FG -linear crossed homomorphism which is bijective. Proof. In view of Proposition I.8, we only have to recall that κG satis?es the Maurer–Cartan equation.

On the cohomology of vector ?elds on parallelizable manifolds

7

II. Cocycles with values in di?erential forms

Let M be a smooth paracompact N -dimensional manifold. In this section we ?rst explain k how a?ne connections can be used to de?ne for each k ∈ N cocycles Ψk ∈ Zc (VM , ?k ). M If, in addition, the tangent bundle of M is trivial, we obtain two more families of cocycles k?1 k 2k?1 Ψk ∈ Zc (VM , ?M ) and Φk ∈ Zc (VM , FM ). These cocycles satisfy the relation d ? Ψk = Ψk p 2 that will play a crucial role in our determination of the spaces Hc (VM , ?M ) in Section IV. The set of a?ne connections ? on M is an a?ne space whose tangent space is the space Γ(T (1,2) (M )) of tensors of type (2, 1). The elements of this space are FM -bilinear maps VM × VM → VM and any such map can also be considered as an FM -linear map VM → Γ(End(T M )), i.e., as a 1 -form with values in the endomorphism bundle End(T M ) ? = T (1,1) (M ) of the tangent bundle T (M ). In this sense we identify Γ(T (1,2) (M )) with the space ?1 (M, End(T M )) of 1 -forms with values in the endomorphism bundle End(T M ). Note that this space carries a natural module structure for the Lie algebra VM , given by the Lie derivative. Lemma II.1. ([Ko74]) Any a?ne connection ? on M de?nes a 1 -cocycle ζ: VM → ?1 (M, End(T M )), where (LX ?)(Y )(Z) := [X, ?Y Z] ? ?[X,Y ] Z ? ?Y [X, Z]. For any other a?ne connection ?′ the corresponding cocycle ζ ′ has the same cohomology class. To understand the cocycle ζ associated to an a?ne connection, we ?rst describe it in a local chart. Remark II.2. If U is an open subset of the vector space V , then any a?ne connection on U is given by ?X Y = dY · X + Γ(X, Y ) with a (2, 1)-tensor Γ. We then have LX (? ? Γ)(Y, Z) = [X, dZ(Y )] ? dZ([X, Y ]) ? d[X, Z](Y ) = d2 Z(X, Y ) + dZ(dY (X)) ? dX(dZ(Y )) ? dZ(dY (X)) + dZ(dX(Y )) ? d(dZ(X) ? dX(Z))(Y ) = d2 Z(X, Y ) ? dX(dZ(Y )) + dZ(dX(Y )) ? d2 Z(Y, X) ? dZ(dX(Y )) + d2 X(Y, Z) + dX(dZ(Y )) = d2 X(Y, Z). This means that LX ? = d2 X + LX Γ. Since the Lie derivative of any symmetric tensor is symmetric, we see that LX ? is symmetric if and only if LX Γ is symmetric, which is the case if Γ is symmetric, and this in turn means that ? is torsion free. As a consequence of the preceding discussion, we can associate to any smooth manifold 1 a canonical cohomology class [ζ] ∈ Hc (VM , ?1 (M, End(T M ))) (cf. [Ko74]). That this class is always non-zero can be seen in local coordinates by showing that there is no (1, 2)-tensor Γ with d2 X = LX Γ for all smooth vector ?elds on a 0 -neighborhood in RN . For constant vector ?elds X , the preceding relation means that Γ is constant and for linear vector ?elds X we see that Γ: RN ? RN → RN should be GLN (R)-equivariant, so that λ2 Γ(v, w) = Γ(λv, λw) = λΓ(v, w) for each λ ∈ R yields Γ = 0 . X → LX ?,

8

Yuly Billig, Karl-Hermann Neeb

Example II.3. Let V := RN and assume that M is parallelizable of dimension N . Then there exists some κ ∈ ?1 (M, V ) such that each κm is invertible, i.e., κ de?nes a trivialization of the tangent bundle of M via M × V → T M, (m, v) → κ?1 v . Then, for each X ∈ VM , m LX κ also in an element of ?1 (M, V ), and since all the linear maps κm are invertible, it can be written as LX .κ = ?θ(X) · κ for some smooth function θ(X) ∈ C ∞ (M, gl(V )). Clearly, C ∞ (M, gl(V )) ? C ∞ (M, R) ? gl(V ) is a Lie algebra with respect to the pointwise bracket and = VM acts naturally by derivations. We claim that θ is a crossed homomorphism: θ([X, Y ]) · κ = ?L[X,Y ] κ = ?LX (LY κ) + LY (LX κ) = LX (θ(Y ) · κ) ? LY (θ(X) · κ) = (LX θ(Y ) ? LY θ(X))κ + θ(Y )LX κ ? θ(X)LY κ = (LX θ(Y ) ? LY θ(X))κ + (θ(Y )θ(X) ? θ(X)θ(Y ))κ = (LX θ(Y ) ? LY θ(X) + [θ(X), θ(Y )])κ. Now let ?X Y := κ?1 (X.κ(Y )) denote the corresponding a?ne connection. Then (2.1) κ(?X Y ) = X.κ(Y ) = ?θ(X)κ(Y ) + κ([X, Y ])

is an immediate consequence of the de?nition of θ . Moreover, the map κ: Γ(End(T M )) → C ∞ (M, gl(V )), satis?es (2.2) In fact, we have κ((LX ?)(Y )(Z)) = κ([X, ?Y Z] ? ?[X,Y ] Z ? ?Y [X, Z]) = κ(?X ?Y Z) + θ(X)κ(?Y Z) ? [X, Y ].κ(Z) ? Y.κ([X, Z]) = XY.κ(Z) + θ(X)(Y.κ(Z)) ? [X, Y ].κ(Z) ? Y.(X.κ(Z) + θ(X) · κ(Z)) = θ(X)(Y.κ(Z)) ? Y.(θ(X).κ(Z)) = ?Y.(θ(X)) · κ(Z) = ?dθ(X)(Y ) · κ(Z), and this calculation shows that κ((LX ?)(Y )) = ?dθ(X)(Y ) for each vector ?eld Y , which is (2.2). Remark II.4. For any a?ne connection ?, the operator η(X) := ?X ? ad X on VM is FM -linear, hence de?nes a section of End(T M ). We thus obtain a map η: VM → Γ(End(T M )), X → ?X ? ad X. κ ? LX ? = ?dθ(X) ∈ ?1 (M, gl(V )). κ(?)(m) = κm ? ?m ? κ?1 m

The space Γ(End(T M )) carries a natural associative algebra structure, hence in particular the structure of a Lie algebra (the gauge Lie algebra of the vector bundle T M ) and VM acts via the Lie derivative by derivations. The map η is a crossed homomorphism for this structure if and only if X → ?X de?nes a representation of VM on itself, i.e., if ? is a ?at connection (Remark I.4). This is in particular the case if ? is obtained from a trivialization κ ∈ ?1 (M, V ) of the tangent bundle. In the latter case, (2.1) implies that κ(η(X)) = ?θ(X).

On the cohomology of vector ?elds on parallelizable manifolds

9

In view of κ(?X ) = LX , we now see that κ(ad X) = κ(LX ) = LX + θ(X), which is a representation of VM on C ∞ (M, V ) if θ is a crossed homomorphism (Remark I.4). We likewise see for the action of VM on Γ(End(T M )) and C ∞ (M, gl(V )) that κ(?) := κ ? ? ? κ?1 leads to κ ? LX = (LX + ad(θ(X)) ? κ, i.e., the representation of VM on Γ(End(T M )) by the Lie derivative is transformed by κ into the representation given by LX + ad(θ(X)) on C ∞ (M, gl(V )). Similarly, the map κ1 : ?1 (M, End(T M )) → ?1 (M, gl(V )), ω →κ?ω

intertwines the Lie derivative on the left hand side with the representation on the right hand side given by LX + ad(θ(X)). De?nition II.5. Next we use the cocycle ζ , associated to the a?ne connection ?, to de?ne k k -cocycles Ψk ∈ Zc (VM , ?k ) depending on second order partial derivatives of vector ?elds. M Let V := RN . For each p ∈ N, we have a polynomial of degree p on gl(V ), invariant under conjugation, given by A → Tr(Ap ). The corresponding invariant symmetric p-linear map is given by β(A1 , . . . , Ap ) = Tr(Aσ(1) · · · Aσ(p) ),

σ∈Sp

and we consider it as a linear GL(V )-equivariant map End(V )?p → R, where GL(V ) acts trivially on R. This GL(V )-equivariant map leads to a vector bundle map βM : End(T M )?p → M × R, where M × R stands for the trivial vector bundle with ?ber R. On the level of bundle-valued di?erential forms, this in turn yields an alternating p-linear map

1 βM : ?1 (M, End(T M ))p → ?p (M, R) = ?p . M

To see that this map is VM -equivariant, we note that in local coordinates we have on an open subset U ? V ? RN the corresponding linear map = ?1 (U, End(T U ))?p ? ?1 (U, R)?p ? End(V )?p → ?p (U, R), = acting on ?1 (U, R)?p as the p-fold exterior product and on End(V )?p as β . If ?: U1 → U2 is a local di?eomorphism, and α ∈ ?1 (U2 , End(T (U2 ))), then ?? α ∈ ?1 (U1 , End(T (U1 ))) is given by (?? α)m (v) := Tm (?)?1 ? α?(m) (Tm (?)v) ? Tm (?).

1 From this it follows that βM is equivariant under di?eomorphisms, and hence with respect to the in?nitesimal action of vector ?elds. 1 We conclude that we can use βM to multiply Lie algebra cocycles (cf. [Fu86], App. F in [Ne04]). In particular, this leads with the cocycle ζ from Lemma II.1 for each k ∈ N to a Lie algebra cocycle k Ψk ∈ Zc (VM , ?k ), M

de?ned by

1 Ψk (X1 , . . . , Xk ) := (?1)k βM (ζ(Xσ(1) ), . . . , ζ(Xσ(k) ))

(cf. [Fu86]). For any other a?ne connection ?′ , the corresponding cocycle ζ ′ , and the associated cocycles Ψ′ , we ?rst note that the di?erence ζ ′ ? ζ is a coboundary, and since products of k cocycles and coboundaries are coboundaries, Ψ′ ? Ψk is a coboundary. Hence its cohomology k k class in Hc (VM , ?k ) does not depend on the choice of the a?ne connection. M

10

Yuly Billig, Karl-Hermann Neeb

Remark II.6. If the connection ? is de?ned by a trivialization of T M as in Example II.3, then ζ(X) is transformed into ?dθ(X) ∈ ?1 (M, gl(V )), which leads to Ψk (X1 , . . . , Xk ) =

σ∈Sk

sgn(σ) Tr dθ(Xσ(1) ) ∧ . . . ∧ dθ(Xσ(k) ) .

Next we construct several equivariant cocycles on a gauge Lie algebra. Theorem II.7. Let g be a ?nite-dimensional Lie algebra, FM ? g be the corresponding gauge Lie algebra, consider ?p as a trivial module of FM ? g, and let ρ: g → gl(V ) be a ?niteM dimensional representation. (a) By FM -linear extension, we get a morphism of cochain complexes, C ? (g, R) → ? Ceq (FM ? g, FM ), where equivariance refers to the action of VM , which gives a homomorphism of cohomology algebras

? H ? (g, R) → Heq (FM ? g, FM ).

(b) The following expression is a VM -equivariant cocycle with values in k -forms, ψk ∈

k Zeq (FM ? g, ?k ): M

ψk (f1 ? x1 , . . . , fk ? xk ) =

σ∈Sk

Tr ρ(xσ(1) ) · · · ρ(xσ(k) )

df1 ∧ · · · ∧ dfk ,

where f1 , . . . , fk ∈ FM , x1 , . . . , xk ∈ g.

(c) The following expression is an equivariant cocycle with values in ?M , k ≥ 1 :

k?1

ψ k (f1 ? x1 , . . . , fk ? xk ) =

σ∈Sk

Tr ρ(xσ(1) ) · · · ρ(xσ(k) )

[f1 df2 ∧ . . . ∧ dfk ],

where f1 , . . . , fk ∈ FM , x1 , . . . , xk ∈ g. Proof. Veri?cation of part (a) is straightforward. Assertions (b) and (c) easily follow from Proposition A.3, which is ?rst used with ?p = ?p to obtain the cocycle property on FM ?gl(V ), M and then we pull it back by the homomorphism idFM ?ρ: FM ? g → FM ? gl(V ). De?nition II.7. Combining Theorems I.7 and II.7 with Renark II.4, we get for each trivialization of the tangent bundle T (M ) of an N -dimensional manifold M by pulling back with the corresponding crossed homomorphism θ: VM → FM ? gl(V ), V = RN , the following cocycles

2k?1 Φk ∈ Z c (VM , FM ),

Φk (X1 , . . . , X2k?1 ) =

σ∈S2k?1

sgn(σ) Tr θ(Xσ(1) ) · · · θ(Xσ(2k?1) )

k and Ψk ∈ Zc (VM , ?M ), de?ned by

k?1

Ψk (X1 , . . . , Xk ) =

σ∈Sk

sgn(σ)[Tr θ(Xσ(1) )dθ(Xσ(2) ) ∧ . . . ∧ dθ(Xσ(k) ) ].

Note that we have for each k ≥ 1 the relation (2.3) d ? Ψk = Ψk .

On the cohomology of vector ?elds on parallelizable manifolds

11

III. Cohomology of smooth vector ?elds with values in di?erential forms

In this section, we ?rst recall Tsujishita’s Theorem ([Tsu81], Thm. 5.1.6, see also 3.3.4), describing the continuous cohomology of VM with the values in di?erential forms. To explain its statement, we recall that each compact manifold M satis?es Tsujishita’s p ? condition (F): HM is ?nite-dimensional and, for each p ∈ N, the subspace ZM of closed p-forms p in ?M has a closed complement. For compact oriented manifolds, the latter assertion is a direct consequence of the Hodge Decomposition Theorem (cf. [AMR83, Thm. 7.5.3]) and the general case can be reduced to this one via the orientable 2 -sheeted covering. Moreover, Beggs shows in [Beg87] (Theorems 4.4 and 6.6) that for all ?nite-dimensional paracompact smooth manifolds the p space ZM of closed p-forms in ?p has a closed complement, so that condition (F) is equivalent M p ? to all cohomology spaces HM , resp., the cohomology algebra HM , being ?nite-dimensional. The following theorem is a key result of this paper because it provides a bridge between Tsujishita’s results an the cocycles Ψk . It will be proved in Appendix C.

? Theorem III.1. Let M be an N -dimensional manifold for which HM is ?nite-dimensional. p q Then Hc (VM , ?M ) vanishes if p > q and for p, n ∈ N0 , we have n p+n Hc (VM , ?p ) ? Hc (VM , FM ) ? Ep , M =

where Ep := span [Ψ1 ]m1 [Ψ2 ]m2 · · · [Ψp ]mp :

p p jmj = p ? Hc (VM , ?p ). M j=1

In particular, is a free H (VM , FM )-module, generated by the non-zero products of the classes [Ψk ], k = 1, . . . , N .

0 From the preceding theorem, we immediately derive with H 0 (VM , FM ) = HM ? R: =

? Hc (VM , ?? ) M

?

Corollary III.2. (a) If p < k , then H p (VM , ?k ) = 0 . M p (b) If M is connected, then H p (VM , ?M ) ? Ep . = Lemma III.3. Let M be a compact connected orientable manifold and ? a volume form on M . Then div: VM → FM , LX ? = div(X)?

1 de?nes a continuous Lie algebra cocycle and furthermore, each closed 1 -form α ∈ ZM de?nes a Lie algebra cocycle VM → FM , X → α(X). These cocycles span the cohomology space 1 Hc (VM , FM ) ? HM ⊕ R[div]. = 1

If, in addition, the volume form is de?ned by a trivialization of the tangent bundle T (M ), then ? div = Φ1 = Ψ1 . Proof. In view of [Fu86, Th. II.4.11], we only have to verify the last part (cf. also [FL80] for the local cohomology of VM with values in ?p ). So let us assume that the volume form can be M written as ? = κ1 ∧ . . . ∧ κN , where the κi are the components of a trivializing 1 -form κ ∈ ?1 (M, RN ). From LX κ = ?θ(X)κ we derive LX κj = ? N θ(X)ji κi and further i=1

N N

LX ? =

i=1

κ1 ∧ . . . ∧ LX κi ∧ . . . ∧ κN = ?

i=1

θ(X)ii κ1 ∧ . . . ∧ κN = ? Tr(θ(X)) · ?.

This proves that div X = ?Φ1 (X) = ?Ψ1 (X).

12

Yuly Billig, Karl-Hermann Neeb

Remark III.4. We take a closer look at the second cohomology of the VM -modules ?p . M 2 From Corollary III.2(a) we know that Hc (VM , ?p ) vanishes for p > 2 , and for p = 2 the M space 2 Hc (VM , ?2 ) = span{[Ψ2 ], [Ψ2 ]} M 1 is 2 -dimensional. For p = 1 we also know that

2 1 1 Hc (VM , ?1 ) = Hc (VM , FM ).[Ψ1 ] = HM · [Ψ1 ] ⊕ R[Ψ1 ∧ Ψ1 ] M 2 is of dimension b1 (M ) + 1 (Lemma III.3). The most intricate case is p = 0 , i.e., Hc (VM , FM ) (cf. Theorem IV.8 below.

Example III.5. Let M = G be an abelian Lie group of dimension N and κ1 , . . . , κN a basis of the space of invariant R-valued 1 -forms on G. Then κ := (κ1 , . . . , κN ) is a closed trivializing 1 form (the Maurer–Cartan form of G). Let X1 , . . . , XN ∈ V(G) be the dual basis of left invariant N vector ?elds. Writing X ∈ V(G) as X = i=1 fi Xi , we obtain LX κi = d(iX κi ) = dfi . Therefore θ(X) ∈ C ∞ (G, glN (R)) is given by θ(X)ij = ?dfi (Xj ), i.e., ?θ(X) is the Jacobian matrix of X with respect to the basis X1 , . . . , XN .

IV. Cohomology with values in di?erential forms modulo exact forms

k HM

Let M be a parallelizable connected compact manifold of dimension N . Since the subspace k of ?M is a trivial VM -module, we derive from Corollary D.5 that

k m Hc (VM , HM ) = 0

(4.1)

for

0 < k ≤ N, m ∈ N0 .

Therefore the short exact sequence of VM -modules

m+1 m 0 → HM → ?M ? →BM → 0 ? m d

induces a long exact sequence in cohomology, hence leads to isomorphisms (4.2a)

m+1 k k d? : Hc (VM , ?M ) → Hc (VM , BM ) m

for

m ∈ N0 , 0 < k ≤ N ? 1

and an exact sequence

m d m+1 N N N m m (4.2b) 0 → Hc (VM , ?M )? ? →Hc (VM , BM ) → Hc +1 (VM , HM ) ? Hc +1 (VM , R) ? HM . ?? ? = N

In particular, we get

m+1 N N dim Hc (VM , ?M ) ≤ dim Hc (VM , BM ).

m

Lemma IV.1. ([Ne06a, Lemma 23], [HS53]) For any smoothly paracompact manifold M we have p p p 0 Hc (VM , ?M ) = (?M )VM = HM . Lemma IV.2. If M is parallelizable, then for each k ∈ N the exterior di?erential d induces a surjective map k?1 k k d? : Hc (VM , ?M ) → Hc (VM , ?k ), [α] → [d ? α], M and the natural map

k k k Hc (VM , BM ) → Hc (VM , ?k ) M

is surjective.

On the cohomology of vector ?elds on parallelizable manifolds

13

Proof. First we recall from De?nition II.7 that d ? Ψk = Ψk . Next we observe that the exterior product of di?erential forms induces maps

p q p+q ZM × BM → BM ,

(α, β) → α ∧ β, (α, [β]) → [α ∧ β].

and hence maps

p Z M × ?M → ?M ,

q

p+q

j 1 1 Since Ψj has values in BM ∧ . . . ∧ BM ? BM , each product Ψj1 ∧ . . . ∧ Ψjr has values in j1 +...+jr , so that BM Ψj1 ∧ Ψj2 ∧ . . . ∧ Ψjr

is a well-de?ned cocycle with values in ?M

j1 +...+jr

, satisfying

d ? Ψj1 ∧ Ψj2 ∧ . . . ∧ Ψjr = (d ? Ψj1 ) ∧ Ψj2 ∧ . . . ∧ Ψjr = Ψj1 ∧ Ψj2 ∧ . . . ∧ Ψjr . For j1 + . . . + jr = k , this implies that the image of d contains all products [Ψj1 ∧ Ψj2 ∧ . . . ∧ Ψjr ] k with ji = k , and, in view of Theorem III.1, these products span Hc (VM , ?k ). M This proves the ?rst part, and the second part of the assertion is an immediate consequence of the ?rst one. In view of (4.3)

q Hc (VM , ?p ) = 0 M

for

q<p

(Corollary III.2(a)), the long exact cohomology sequence associated to the short exact sequence

p 0 → BM → ?p → ?M → 0 M p

(which splits topologically by condition (F)), leads to isomorphisms (4.4a)

p p q Hc (VM , ?M ) ? Hc (VM , BM ) = q+1

for

q < p ? 1.

For q = p ? 1 , Lemma IV.2 leads to an exact sequence (4.4b)

p p p?1 p?1 → p 0 = Hc (VM , ?p ) → Hc (VM , ?M ) → Hc (VM , BM ) → Hc (VM , ?p ) → 0. M M p

From [Ne06a, Prop. 6], we recall: Lemma IV.3. For each closed (p + q)-form ω ∈ ?p+q , the prescription M ω [p] (X1 , . . . , Xp ) := [iXp . . . iX1 ω] ∈ ?M

p de?nes a continuous p-cocycle in Zc (VM , ?M ). q q

2 Calculating Hc (VM , ?M )

m

In this subsection we address the problem to determine the second cohomology space m 2 Hc (VM , ?M ) for a parallelizable manifold M of dimension N . We start by observing that we have the following isomorphisms

m+1 2 Hc (VM , ? ) ? Hc (VM , BM ) = 2 m

for N ≥ 3 for m > 1

by by by by

(4.2a) (4.4a) (4.2a) (4.4a)

(4.5)

? H 1 (VM , ?m+1 ) = c M ? H 1 (VM , B m+2 ) =

c

? = = We thus obtain

for N ≥ 2 M m+2 0 Hc (VM , ?M ) for each m m+2 HM by Lemma IV.1.

14 Proposition IV.4.

Yuly Billig, Karl-Hermann Neeb

(a) For N ≥ 3 and m ≥ 2 the map

m+2 2 HM → Hc (VM , ?M ), m

[ω] → [ω [2] ]

is an isomorphism. (b) For N ≥ 2 and m ≥ 1 the map

m+1 1 HM → Hc (VM , ?M ), m

[ω] → [ω [1] ]

is an isomorphism. Proof. In view of (4.5), we only have to see that the isomorphisms are implemented by the maps ω → ω [2] , resp., ω → ω [1] . m+2 (a) We start with a closed (m + 2)-form ω , representing an element of HM , which m+2 m+2 we consider a VM -?xed element in ?M . A corresponding 1 -cocycle with values in BM is [1] given by α(X) := LX ω = d(iX ω) = (d ? ω )(X). This already shows that the corresponding m+1 ?M -valued 1 -cocycle is ω [1] (cf. Lemma IV.3). To distinguish it from the exterior di?erential d, we write dV(M) for the Chevalley– m+1 Eilenberg di?erential on C ? (VM , ?? ). To see the corresponding 2 -cocycle with values in BM , M [1] we recall from [Ne06a, Lemma 5] that for ω (X) := iX ω we have (4.6) (dVM ω [1] )(X, Y ) = d(iX iY ω) = ?(d ? ω [2] )(Y, X).

m 2 Hence ?ω [2] ∈ Zc (VM , ?M ) is the 2 -cocycle corresponding to [ω] under the isomorphisms m m+2 ? 2 HM = Hc (VM , ? ) in (4.5). (b) is proved as in (a).

Now we turn to the cases not covered by Proposition IV.4. The following proposition was the original motivation for the present paper: Theorem IV.5. For N ≥ 2 we have

1 2 Hc (VM , ?M ) ? HM ⊕ R[Ψ1 ∧ Ψ1 ] ⊕ R[Ψ2 ], = 3 3 where HM embeds via the map [ω] → [ω [2] ]. 1 2 Proof. First we assume that N ≥ 3 . For m = 1 , we ?rst get from (4.2a) that Hc (VM , ?M ) ? = 2 2 Hc (VM , BM ) and further from (4.4b) the exact sequence 2 2 2 1 0 → Hc (VM , ?M ) ?→ Hc (VM , BM ) → Hc (VM , ?2 ) = R[Ψ2 ] ⊕ R[Ψ2 ], M 1 2

where the inclusion on the left maps [ω [1] ] to [ω [2] ] (cf. the proof of Proposition IV.4). Since we know from Proposition IV.4 that the map

3 1 HM → Hc (VM , ?M ), 2

[ω] → [ω [1] ]

2 is an isomorphism, the assertion follows because the classes [Ψ2 ] and [Ψ2 ] in Hc (VM , ?2 ) are 1 M 2 2 contained in the image of Hc (VM , BM ) (Lemma IV.2) and satisfy

[d ? (Ψ1 ∧ Ψ1 )] = [Ψ2 ] 1

1 1 2 Hc (VM , ?M ) ? Hc (VM , BM ) ? HM = 1 = 2

and

[d ? Ψ2 ] = [Ψ2 ].

Now we consider the case N = 2 . For m = 1 we get from (4.2a) and Proposition IV.4 and

2 2 3 1 Hc (VM , ?M ) ? Hc (VM , BM ) = {0}. = 1

2 Therefore the exact sequence BM ?→ ?2 → ?M leads to an exact sequence M 2 2 2 2 1 0 = Hc (VM , ?M ) → Hc (VM , BM ) → Hc (VM , ?2 ) → Hc (VM , ?M ) = 0, M 2 2

showing that the map

2 2 2 Hc (VM , BM ) → Hc (VM , ?2 ) ? R[Ψ2 ] ⊕ R[Ψ2 ] M = 1

is an isomorphism. On the other hand, (4.2b) yields an embedding

1 2 2 2 d?: Hc (VM , ?M ) ?→ Hc (VM , BM ) ? Hc (VM , ?2 ) ? R[Ψ2 ] ⊕ R[Ψ2 ], = 2 1 M =

and Lemma IV.2 asserts that this embedding is surjective. This completes the proof.

On the cohomology of vector ?elds on parallelizable manifolds

15

Proposition IV.6.

2 For N = 2 we have Hc (VM , ?M ) = 0 for m ≥ 2 and 2 dim Hc (VM , ?M ) = 2. 1

m

Proof. For N = 2 we have ?m = 0 for m ≥ 3 , so that ?M vanishes in these cases. M 2 2 2 3 For m = 2 we get from (4.2b) an inclusion Hc (VM , ?M ) ?→ Hc (VM , BM ) = 0. Therefore m 2 Hc (VM , ?M ) vanishes for m ≥ 2 . The case m = 1 follows from Theorem IV.5. So far we have covered all cases m ≥ 1 for N ≥ 2 , and we are left with the case m = 0 or N = 1 . First we recall the results for N = 1 : Example IV.7. (a) (cf. [FF01, Th. 30]) For N = 1 and M = S1 the algebra H ? (VS1 , FS1 ) is a free graded commutative algebra generated by two elements α1 , α2 of degree 1 and one element β of degree 2 . In particular

1 Hc (VS1 , FS1 ) = Rα1 ⊕ Rα2 1

m

and

2 Hc (VS1 , FS1 ) = Rα1 α2 ⊕ Rβ.

1 Furthermore ?S1 = HS1 is one-dimensional and a trivial VS1 -module, so that 1 H 2 (VS1 , ?S1 ) = H 2 (VS1 , R) ? R =

(Theorem 29 in [FF01], p. 195). (b) For M = RN the cohomology algebra H ? (VRN , FRN ) ? H ? (glN (R), R) is the free = exterior algebra generated by the classes [Φk ], k = 1, . . . , N (cf. De?nition II.7; [Tsu77], §4). Now we turn to the case m = 0 . Theorem IV.8. For N ≥ 2 the map

2 1 2 HM ⊕ HM → Hc (VM , FM ),

([α], [β]) → [α + β ∧ Ψ1 ]

is a linear isomorphism. Proof. In view of ?M = FM , (4.2a/b) provide an embedding

2 2 1 d? : Hc (VM , FM ) ?→ Hc (VM , BM ), 0

[ω] → [d ? ω]

which is an isomorphism for N > 2 . We also get from Proposition IV.4 for N ≥ 2 an isomorphism: 1 2 1 HM → Hc (VM , ?M ), [ω] → [ω [1] ]. In view of Lemma III.3, we have

2 1 Hc (VM , ?1 ) = Hc (VM , FM ) · [Ψ1 ] ? HM ⊕ R[Ψ1 ∧ Ψ1 ]. = 1 M

We now consider the exact sequence

1 1 1 ? 2 Hc (VM , BM ) → Hc (VM , ?1 )? → 1 M →Hc (VM , ?M ) = HM 2 1 2 2 ? ? ?→Hc (VM , BM )? →Hc (VM , ?1 )? →Hc (VM , ?M ), ?? ? M ? ?d f g 1 0 1

in which we know all terms except the middle one, which we are interested in. The one1 1 dimensional space Hc (VM , ?1 ) is generated by [Ψ1 ], which has values in BM , so that its M 1 image in ?M vanishes. This yields the 0 -arrow in the upper row of the diagram (Lemma IV.2). 1 2 1 1 To calculate the connecting map Hc (VM , ?M ) ? HM → Hc (VM , BM ), we recall from (4.6) that = 2 we have for each closed 2 -form ω the relation δ([ω [1] ]) = [dVM ω [1] ] = ?[d ? ω [2] ] = ?d? [ω].

16

Yuly Billig, Karl-Hermann Neeb

Let β be a closed 1 -form on M , considered as a 1 -cocycle VM → FM . Then

2 β ∧ Ψ1 ∈ Zc (VM , FM )

satis?es d ? (β ∧ Ψ1 ) = (d ? β) ∧ Ψ1 + β ∧ d ? Ψ1 = (d ? β) ∧ Ψ1 + β ∧ Ψ1 . The 1 -cochain γ: VM → ?1 , M satis?es dV(M) γ = (dV(M) Ψ1 ) ∧ β ? Ψ1 ∧ (dV(M) β) = ?Ψ1 ∧ dV(M) β. Since the Cartan formula implies that (dV(M) β)(X) = LX β = iX (dβ) + d(iX β) = d(β(X)),

2 we obtain dV(M) γ = ?Ψ1 ∧ (d ? β), showing that (d ? β) ∧ Ψ1 ∈ Bc (VM , ?1 ). This implies that M

γ(X) := Ψ1 (X) · β = (Ψ1 · β)(X)

d? [β · Ψ1 ] = [d ? (β ∧ Ψ1 )] = [β ∧ Ψ1 ]. We also recall that

2 1 Hc (VM , ?1 ) = (HM ∧ [Ψ1 ]) ⊕ R(Ψ1 ∧ Ψ1 ) ? HM ⊕ R. = 1 M 1 From the preceding considerations, we see that the subspace HM ∧ [Ψ1 ] lies in the range of f . 2 Since g([Ψ1 ∧ Ψ1 ]) = [Ψ1 ] is non-zero, we derive 1 im(f ) = ker(g) = HM ∧ [Ψ1 ]. 1 1 From the exactness in Hc (VM , BM ) we now see that the natural map 2 1 2 1 HM ⊕ HM → Hc (VM , BM ),

([α], [β]) → [d ? α + β ∧ Ψ1 ] = [d ? (α + β · Ψ)]

is a linear isomorphism. As the elements α+β·Ψ form FM -valued 2 -cocycles, we ?nally conclude that the inclusion d? is also surjective for N = 2 . This completes the proof. Problem 1. Determine the spaces H 2 (VM , ?M ) for all connected smooth manifolds M , without assuming that M is parallelizable.

1

Appendix A. Lie algebra cocycles with values in associative algebras

Let A be a unital associative algebra and AL := (A, [·, ·]) be the underlying Lie algebra. We consider A as an AL -module with respect to the adjoint representation x.y := [x, y]. We write mA : A × A → A for the product map and bA : A × A → A for the commutator bracket and observe that both are AL -invariant. We take a closer look at the Lie algebra complex (C ? (AL , A), d), where we use the multiplication on A, which is AL -equivariant, to de?ne a multiplication on C ? (AL , A) by (α ∧ β)(x1 , . . . , xp+q ) := 1 p!q! sgn(σ)mA α(xσ(1) , . . . , xσ(p) ), β(xσ(p+1) , . . . , xσ(p+q) )

σ∈Sp+q

for α ∈ C p (AL , A) and β ∈ C q (AL , A) (cf. [Fu86, Sect. I.3.2]). With Alt(γ)(x1 , . . . , xr ) :=

σ∈Sr

sgn(σ)α(xσ(1) , . . . , xσ(r) ),

On the cohomology of vector ?elds on parallelizable manifolds

17

this means that α∧β =

1 1 Alt(α · β) = Alt(mA ? (α, β)). p!q! p!q!

An easy induction implies that n-fold products in C ? (AL , A) are given by α1 ∧ · · · ∧ αn = 1

n j=1

pj !

Alt(α1 · · · αn )

for

αi ∈ C pi (AL , A).

We thus obtain an associative di?erential graded algebra (C ? (AL , A), d) (cf. [Ne04, App. F]; see also [Tsu81, p. 30]). Note that idA ∈ C 1 (AL , A) is a 1 -cochain with dAL (idA )(x, y) = x.y ? y.x ? [x, y] = [x, y] ? [y, x] ? [x, y] = [x, y], so that dAL (idA ) = bA . It follows in particular that bA is a 2 -coboundary. We therefore obtain a sequences of coboundaries

n?1 bn = dAL (idA ∧bA ) ∈ B 2n (AL , A) ? Z 2n (AL , A). A

Lemma A.1. If T : A → z is a linear map vanishing on all commutators, then T ? bk vanishes A for each k ∈ N, and if z is considered as a trivial AL -module, then

k?1 ?k := T ? (idA ∧bA ) = T ? (idA )2k?1 ∈ Z 2k?1 (AL , z),

is a cocycle given by ?k (x1 , . . . , x2k?1 ) =

σ∈S2k?1

sgn(σ)T (xσ(1) · · · xσ(2k?1) ).

Proof.

Arguing as in Remark I.2 of [Ne06b], we see that for each n ∈ N0 , we have bn = A 1 Alt(bA · · · bA ) = Alt(mA · · · mA ) = Alt(id2n ) = (idA )2n , A 2n bn (x1 , . . . , x2n ) = A

σ∈S2n

i.e., sgn(σ)xσ(1) · · · xσ(2n) .

Let τ be the cyclic permutation τ = (1 2 . . . 2n). Then sgn(τ ) = ?1 , but, for each σ ∈ S2n , we have T (xστ (1) · · · xστ (2n) ) = T (xσ(2) · · · xσ(2n) xσ(1) ) = T (xσ(1) xσ(2) · · · xσ(2n) ). This leads to T ? bn = ?TA ? bn , which implies that T ? bn vanishes. A A A Since bA is a cocycle, we now obtain

k?1 k?1 d?k = d(T ? (idA ∧bA )) = T ? (d(idA ) ∧ bA ) = T ? (bk ) = 0. A

The following theorem describes an important application of the preceding construction, namely that for A = gln (R) we thus obtain a set of generators of the cohomology algebra ([Fu86, Th. 2.1.6]): Theorem A.2. The cohomology algebra H ? (glN (R), R) is generated by the cohomology classes of the cocycles ?k ∈ Z 2k?1 (glN (R), R), k = 1, . . . , N, given by ?k (x1 , . . . , x2k?1 ) =

σ∈S2k?1

sgn(σ) Tr xσ(1) . . . xσ(2k?1)

for

x1 , . . . , x2k?1 ∈ glN (R).

18

Yuly Billig, Karl-Hermann Neeb

Lie algebra cocycles with values in di?erential forms Let (?? , d) be a di?erential graded algebra whose multiplication is denoted α ∧ β , V a ?nite-dimensional vector space, and put A := ?0 ? End(V ) and B := ?? ? End(V ). Then B also carries the structure of a di?erential graded algebra with di?erential dB := d ? idEnd(V ) which is not graded commutative. Note that AL ? ?0 ? gl(V ) as Lie algebras. = Proposition A.3. For each k ∈ N, we obtain cocycles ψk ∈ Z k (AL , ?k ) and ψ k ∈ k k?1 k?2 Z (AL , ? /d? ) satisfying d ? ψ k = ψk by ψk (f1 ? x1 , . . . , fk ? xk ) :=

σ∈Sk

Tr(xσ(1) · · · xσ(k) )df1 ∧ . . . ∧ dfk

and ψ k (f1 ? x1 , . . . , fk ? xk ) :=

σ∈Sk

Tr(xσ(1) · · · xσ(k) )[f1 · df2 ∧ . . . ∧ dfk ] ∈ ?k?1 /d?k?2 .

Proof.

We ?rst consider the linear map ? = dB : AL → ?1 ? End(V ), f ? x → df ? x

and note that (dAL ?)(a, b) = a.db ? b.da ? d[a, b] = [a, db] ? [b, da] ? [da, b] ? [a, db] = 0 implies that ? is a 1 -cocycle. This implies that its ∧-powers ?k ∈ Z k (AL , ?k ? End(V )) are k -cocycles and hence that ψk = Tr ??k ∈ Z k (AL , ?k ), because Tr: ?k ? End(V ) → ?k is a morphism of the AL -module ?k ? End(V ) onto the trivial module ?k . That ψ k is alternating follows from [f1 ∧ df2 ∧ . . . ∧ dfk ] + [f2 ∧ df1 ∧ df3 ∧ . . . ∧ dfk ] = [d(f1 f2 ) ∧ . . . ∧ dfk ] = [d(f1 f2 ∧ df3 ∧ . . . dfk )] = 0. It remains to show that ψ k is a k -cocycle. To this end, we consider idA ∧?k?1 ∈ C k (AL , ?k?1 ? End(V )) and observe that dAL (idA ∧?k?1 ) = dAL (idA ) ∧ ?k?1 = bA ∧ ?k?1 = idA ∧ idA ∧?k?1 . If q: ?k → ?k /d?k?1 , β → [β] is the quotient map, we have q ? Tr ?(idA ∧?k?1 ) (f1 ? x1 , . . . , fk ? xk ) =

σ∈Sk

sgn(σ) Tr(xσ(1) · · · xσ(k) )[fσ(1) dfσ(2) ∧ . . . ∧ dfσ(k) ] Tr(xσ(1) · · · xσ(k) )[f1 df2 ∧ . . . ∧ dfk ] = ψ k (f1 ? x1 , . . . , fk ? xk ).

σ∈Sk

=

To see that ψ k is a cocycle, it therefore su?ces to show that dψ k = q ? Tr ?(idA ∧ idA ∧?k?1 ) = 0.

On the cohomology of vector ?elds on parallelizable manifolds

19

Explicitly, we have q ? Tr ?(id2 ∧?k?1 ) (f1 ? x1 , . . . , fk+1 ? xk+1 ) A =

σ∈Sk+1

sgn(σ) Tr(xσ(1) · · · xσ(k+1) )[fσ(1) fσ(2) dfσ(3) ∧ . . . ∧ dfσ(k+1) ].

If τ = (1 2 . . . k + 1) is the cyclic permutation, we obtain sgn(στ ) Tr(xστ (1) · · · xστ (k+1) )[fστ (1) fστ (2) dfστ (3) ∧ . . . ∧ dfστ (k+1) ] = (?1)k sgn(σ) Tr(xσ(1) · · · xσ(k+1) )[fσ(2) fσ(3) dfσ(4) ∧ . . . ∧ dfσ(k+1) ∧ df1 ] = sgn(σ) Tr(xσ(1) · · · xσ(k+1) )[fσ(2) fσ(3) dfσ(1) ∧ dfσ(4) ∧ . . . ∧ dfσ(k+1) ]. Now the relation [f1 f2 df3 ∧d(· · ·)]+[f2 f3 df1 ∧d(· · ·)]+[f3 f1 df2 ∧d(· · ·)]+ = [d(f1 f2 f3 )∧d(· · ·)] = [d(f1 f2 f3 · · ·)] = 0 implies that 3ψ k is a cocycle, so that ψ k is a cocycle.

Appendix B. Cohomology of formal vector ?elds

In this section, we will review results of Gelfand–Fuks on the cohomology of formal vector ?elds because this is needed to reduce Theorem III.1 to Tsujishita’s result. Fix N ∈ N and write FN = R[[x1 , . . . , xN ]] for the commutative algebra of formal power series in the variables x1 , . . . xN , endowed with the projective limit topology, and let WN be the Lie algebra of continuous derivations of FN :

N

WN =

j=1

FN

? . ?xj

Consider also the subalgebra L0 in WN consisting of vector ?elds that vanish at the origin:

N

L0 =

j=1

fj (x)

? ?xj

:

fj (0) = 0, j = 1, . . . , N ,

satisfying

WN ? L0 ? RN , =

where the second factor corresponds to the abelian Lie subalgebra of constant (formal) vector ?elds. For X ∈ WN , we write J(X) ∈ FN ? glN (R) for the Jacobian of X , de?ned for X =

i ? fi ?xi by J(X)ij = ?fi ?xj

. Then

?J: WN → FN ? glN (R) is a crossed homomorphism whose kernel is the subalgebra of constant vector ?elds. From the relation ?J([X, Y ]) = ?X.J(Y ) + Y.J(X) + [?J(X), ?J(Y )], it follows that ?J0 : L0 → glN (R), X → ?J(X)(0) is a surjective homomorphism of Lie algebras, restricting to an isomorphism on the subalgebra of linear vector ?elds. Let ?p be the space of formal p-forms in N variables, considered as a free FN -module N and also as a WN -module, with respect to the action de?ned by the Lie derivative.

20

Yuly Billig, Karl-Hermann Neeb

De?nition B.1. Using the crossed homomorphism θ := ?J , we get with Lemma A.1, Proposition A.3 and Theorem I.7 the following cocycles

2k?1 ΦW ∈ Z c (WN , FN ), k

ΦW (X1 , . . . , X2k?1 ) = k

σ∈S2k?1

sgn(σ) Tr θ(Xσ(1) ) . . . θ(Xσ(2k?1) ) ,

k ΨW ∈ Zc (WN , ?k ), k N

ΨW (X1 , . . . , Xk ) = k

σ∈Sk

sgn(σ) Tr dθ(Xσ(1) ) ∧ . . . ∧ dθ(Xσ(k) ) ,

k?1 k?2 k and Ψk ∈ Zc (WN , ?N /d?N ), de?ned by

W

Ψk (X1 , . . . , Xk ) =

σ∈Sk

W

sgn(σ)[Tr θ(Xσ(1) )dθ(Xσ(2) ) ∧ . . . ∧ dθ(Xσ(k) ) ].

The following theorem is a re-formulation of the results of Gelfand-Fuks ([GF70a]), describing the cohomology of WN with values in the modules ?p . Let V = RN , considered as the N canonical module of glN (R) and write ev0 : ?p → Λp (V ′ ), N ω → ω(0)

for the evaluation map. We consider V as a module of L0 by pulling back the module structure from glN (R), so that ev0 is a morphism of L0 -modules. Theorem B.2. q p p (a) For each p, q ∈ N0 , the map Cc (WN , ?N ) → Cc (L0 , Λq (V ′ )), ω → ev0 ?ω | L0 induces an isomorphism p p Hc (WN , ?q ) → Hc (L0 , Λq (V ′ )). N

? ? (b) Hc (L0 , Λ? (V ′ )) ? Hc (L0 , R) ? Hc (L0 , glN (R), Λ? (V ′ )) as bigraded algebras. = ? (c) The inclusions glN (R) ?→ L0 ?→ WN induce isomorphisms of graded algebras ? ? ? Hc (WN , R) → Hc (L0 , R) → Hc (glN (R), R). ? In particular, Hc (WN , R) is an exterior algebra with generators of degree 2k ? 1 , k = 1, . . . , N . ? k (d) Let ΨL := ev0 ?ΨW | L0 ∈ Zc (L0 , glN (R), Λk (V ′ )). Then Hc (L0 , glN (R), Λ? (V ′ )) is a k k quotient of the commutative algebra generated by the cohomology classes [ΨL ], k = 1, . . . , N , k by the ideal spanned by the elements of degree exceeding N .

Proof. We only explain how this can be derived from [Fu86]. (a) Write U (g) for the enveloping algebra of a Lie algebra g. First we observe that the map ζ: ?p → HomL0 (U (WN ), Λp (V ′ )) ? Hom(S(RN ), Λp (V ′ )) ? FN ? Λp (V ′ ), = = N ζ(α)(D) := (D.α)(0) is an isomorphism of WN -modules. Hence ?p is coinduced, as a WN -module, from the L0 N module Λp (V ′ ). Note that, since L0 is of ?nite codimensional in WN , no problems arise from continuity requirements. Therefore (a) follows from a general result on coinduced modules ([Fu86, Th. 1.5.4]). (b) The proof of part (b) is based on the Hochschild–Serre Spectral Sequence associated with the ?ltration on C ? (L0 , Λ? (V ′ )) relative to the subalgebra glN (R) ? L0 . By Theorem p,0 1.5.1(ii) in [Fu86], H p (L0 , Λ? (V ′ )) is the E2 -term in the spectral sequence, while in the proof p,0 p,0 p,0 ′ of Theorem 2.2.7 in [Fu86], it is shown that E2 = E1 , and the term E1 is calculated explicitly. (c), (d) follow from (a), combined with [Fu86, Thms. 2.2.7 and 2.2.7’].

On the cohomology of vector ?elds on parallelizable manifolds

21

From the formulas in [Fu86] it is not completely obvious that our formula for ΨL describes k the same cocycle (up to a scalar factor). Fuks describes ΨL as an element of k Λk (V ′ ) ? Λk (S 2 (V ) ? V ′ ).

k In our context, the cocycle ΨL ∈ Zc (L0 , glN (R), Λk (V ′ )) is given by the formula k

ΨL (X1 , . . . , Xk ) = (?1)k k

σ∈Sk

sgn(σ) Tr dJ(Xσ(1) )(0) ∧ . . . ∧ dJ(Xσ(k) )(0) ∈ Λk (V ′ ).

For each X ∈ L0 , the constant term of the 1 -form dJ(X) ∈ ?1 ? glN (R) corresponds to the N quadratic term in X , which can be identi?ed with an element of S 2 (V ′ ) ? V ? Sym2 (V ) ? V . = For the basis element Xi1 ,i2 ,? := xi1 xi2 ?? , we have, in terms of the matrix basis Eij of glN (R): J(X) = xi1 E?i2 + xi2 E?i1 This leads to (?1)k ΨL (Xi11 ,i12 ,?1 , . . . , Xik1 ,ik2 ,?k ) k =

σ∈Sk

τi ∈S2 i=1,...,k

and

d(J(X)) = dxi1 E?i2 + dxi2 E?i1 =

τ ∈S2

dxiτ (1) E?iτ (2) .

Tr dxiσ(1),τ1 (1) E?σ(1) iσ(1)τ1 (2) ∧ · · · ∧ dxiσ(k),τk (1) E?σ(k) iσ(k)τk (2) δiσ(1)τ1 (2) ,?σ(2) δiσ(2)τ2 (2) ,?σ(3) · · · δiσ(k)τk (2) ,?σ(1) · dxiσ(1),τ1 (1) ∧ · · · ∧ dxiσ(k),τk (1) .

σ∈Sk

τi ∈S2 i=1,...,k

=

From this formula it is not hard to verify that our ΨL are multiples of those in [Fu86]. k

Appendix C. Higher frame bundles and di?erential forms

In this appendix we shall prove our key Theorem III.1. The major part of the work consists in explaining why Theorem III.1 can be derived from Tsujishita’s [Tsu81]. For that we need the passage from the cohomology of formal vector ?elds to the cohomology of VM and ?nally to link it to Theorem III.1. Let M be an N -dimensional smooth manifold. For k ∈ N0 ∪ {∞} we write J k (M ) for its k -frame bundle whose elements are k -jets [α] of local di?eomorphism α: (RN , 0) → M . Then J 0 (M ) = M and J 1 (M ) is the frame bundle of M whose elements are the isomorphisms RN → Tm (M ), m ∈ M . Evaluating [α] in 0 leads to a natural map J k (M ) → M which exhibits k J k (M ) as a ?ber bundle over M . We write Jm (M ) for the ?ber over m ∈ M . For ? ≤ k we have projection maps ? π?,k : J k (M ) → J ? (M ), [α] → j0 (α). In particular, we have π0,k ([α]) = α(m) and π1,k ([α]) = T0 (α). Let Gk denote the group of k -jets of local di?eomorphisms of RN ?xing 0 . As a set, Gk is the set of all polynomial maps ?: RN → RN of degree ≤ k without constant term for which d?(0) is invertible. The multiplication is given by

k ?1 ?2 := j0 (?1 ? ?2 ).

For ? ≤ k we have natural surjective homomorphisms q?,k : Gk → G? ,

? α → j0 (α),

22

Yuly Billig, Karl-Hermann Neeb

cutting o? all terms of order > ? . We put Gk := ker(qk,∞ ) and note that (C.1) in a natural way. For k < ∞, the group Gk is a ?nite-dimensional Lie group, G1 ? GLN (R), and = Gk ? Gk ? GLN (R), = 1 where Gk := ker(q1,k ) is a simply connected nilpotent Lie group. Identifying G∞ with the 1 projective limit of all groups Gk ? G∞ /Gk , we obtain a topology for which it actually is a Lie = group with Lie algebra L(G∞ ) ? L0 and Lk := L(Gk ) is a ?nite-codimensional ideal of L0 (cf. = Appendix B). The normal subgroup G1 is pro-nilpotent, its exponential function expG1 : L1 → G1 is a di?eomorphism, and (C.1) is a semidirect product of Lie groups. The group Gk acts on J k (M ) from the right by [α].? := [α ? ?]. This action is transitive on the ?bers and de?nes on J k (M ) the structure of a smooth Gk -principal bundle. The group Di?(M ) acts on each frame bundle J k (M ) by ?.[α] := [? ? α], and the corresponding homomorphism of Lie algebras is given by (C.2) γ k : VM → VJ k (M) , γ k (X)[α] = d dt [FlX ? α]. t G∞ ? G1 ? GLN (R) = G∞

t=0

Since the action of Di?(M ) on J k (M ) commutes with the action of the structure group Gk , (C.3)

G γ k (VM ) ? VJ k (M) .

k

Lemma C.1. ([Tsu81, Lemma 4.2.2]) Let k ∈ N ∪ {∞} , ? be an a?ne connection on M , and D ? T M the open domain of the corresponding exponential function Exp? : D → M , which is an open neighborhood of the zero section. Then s: J 1 (M ) → J k (M ), is a smooth section. Note that for each m ∈ M the intersection Dm := D ∩ Tm (M ) is an open zero neighborhood and that Exp? : Dm → M is a smooth map with T0 (Exp? ) = idTm (M) , hence a local m m di?eomorphism. Now the map J 1 (M ) × Gk → J k (M ), (α, g) → s(α)g is a di?eomorphism and 1 the map F : J k (M ) → Gk , s(α)g → g, α ∈ J 1 (M ), 1 is smooth and Gk -equivariant. For g ∈ GLN (R) we have s(α ? g) = s(α).g , which implies that 1 (C.4) F (α.g) = g ?1 F (α)g for all α ∈ J k (M ).

k s(α) := j0 (Exp? ?α) m

Hence F is equivariant with respect to the action of Gk ? Gk ? GLN (R) on Gk from the right = 1 1 ?1 by g.(g1 , g0 ) = g0 gg1 g0 . Since Tsujishita uses a realization of Lie algebra cohomology in terms of right invariant, resp., equivariant di?erential forms on the corresponding group, we brie?y discuss the relevant identi?cations in the following remark.

On the cohomology of vector ?elds on parallelizable manifolds

23

Remark C.2. (a) Let G be a Lie group. A smooth G-module is a topological vector space V on which G acts smoothly by linear maps. We write ρV : G × V → V, (g, v) → ρV (g)(v) =: g.v

for the action map. Further, let M be a smooth manifold on which G acts from the right by M × G → M, (m, g) → m.g =: ρM (m). We call a p-form α ∈ ?p (M, V ) equivariant if we have for each g g ∈ G the relation (ρM )? α = ρV (g)?1 ? α. g We write ?p (M, V )G for the space of G-equivariant p-forms on M . This is the space of G-?xed elements with respect to the action of G on ?p (M, V ), given by g.α := ρV (g) ? (ρM )? α . g (b) For the right action of G on M = G by left multiplication x.g := g ?1 x we obtain the space ?p (G, V ) of left equivariant forms, i.e., forms satisfying ? λ? ω = ρV (g) ? ω, g In [ChE48] it is shown that the evaluation map

? ev1 : (?? (G, V ), d) → (Cc (g, V ), dg ), ?

g ∈ G.

ω → ω1

yields an isomorphism of cochain complexes (cf. [Ne04] for the unproblematic extension to in?nitedimensional Lie groups). ? (c) There is also a realization of the complex (Cc (g, V ), dg ) by right equivariant di?erential ?1 forms on G: If ηG : G → G, g → g is the inversion map, then for each left equivariant p-form ? α ∈ ?p (G, V ) the p-form α := ηG α is right equivariant, i.e., ? ρ? α = ρV (g)?1 ? α g for each g ∈ G.

We thus obtain an isomorphism of cochain complexes

? ηG : (?? (G, V ), d) → (?? (G, V ), d). ? r

Since T1 (ηG ) = ? idG , we also obtain an isomorphism

? ev1 : (?? (G, V ), d) → (Cc (g, V ), dg ), r

ω → (? idg )? ω1 .

For each g ∈ G and ω ∈ ?p (G, V ) the form λ? ω is also right equivariant and satis?es g r (λ??1 ω)1 (x1 , . . . , xp ) = ωg?1 (g ?1 .x1 , . . . , g ?1 .xp ) = ρV (g).ω(Ad(g)?1 .x1 , . . . , Ad(g)?1 .xp ), g showing that ev1 intertwines the action of G by left translation on ?p (G, V ) with the natural r p action of G on the cochain space Cc (g, V ). (d) There is an alternative realization of the complex (C ? (g, V ), dg ) as the space of left, resp., right invariant V -valued di?erential forms on G. First we write the Chevalley–Eilenberg di?erential as dg = d0 + ρV ∧, g where d0 is the di?erential corresponding to the trivial module structure on V and ρV : g → gl(V ) g is the homomorphism of Lie algebras, de?ning the g-module structure on V . If κG ∈ ?1 (G, g) denotes the left Maurer–Cartan form of G, we obtain κV := ρV ? κG ∈ 1 ? (G, gl(V )) and a corresponding covariant di?erential on ?? (M, V ): dκ ω = dω + κV ∧ ω. Since κ is left invariant, this di?erential preserves the subspace of left invariant forms, and it is ? easy to see that evaluation in 1 intertwines it with dg on Cc (g, V ). In a similar fashion, one obtains a realization by right invariant di?erential forms with the appropriate di?erential.

24

Yuly Billig, Karl-Hermann Neeb

Let A be a ?nite-dimensional smooth G∞ -module, where we assume that the action of the non-connected Lie group GLN (R) on A is the one obtained by restricting the action of GLN (C) on the complexi?cation AC , hence completely determined by the action of glN (R) on A. We then have natural identi?cations (C.5) C ? (L0 , gl (R), A) ? C ? (L1 , A)glN (R) ? ?? (G1 , A)GLN (R) = =

c N c r

([Tsu81, Lemma 3.3.4], Remark C1b(c)). Although GLN (R) is not connected, this follows from a complexi?cation argument since GLN (C) is connected. p Further, let ω ∈ Cc (L1 , A)GLN (R) ? Cc (L1 , A)glN (R) and ω ∈ ?p (G1 , A) the corresponding = p r right equivariant A-valued p-form with ω1 = (?1)p ω (cf. Remark C.2(c)). For g1 ∈ G1 we then have ρ?1 F ? ω = (F ? ρg1 )? ω = (ρg1 ? F )? ω = ρA (g1 ) ? F ? ω g and for g0 ∈ GLN (R) we obtain with (C.4)

?1 ρ?0 F ? ω = (F ? ρg0 )? ω = (cg0 ? F )? ω = F ? (c?1 )? ω = ρA (g0 ) ? F ? ω, g0 g

hence the GLN (R)-equivariance of ω . We thus obtain for k = ∞ a morphism of chain complexes (C.6)

? F ? : Cc (L1 , A)GLN (R) → ?? (J ∞ (M ), A)G ,

∞

ω → F ? ω.

The map γA de?ned below is the one used by Tsujishita in [Tsu81, p. 62]. Proposition C.3. Let A := J ∞ (M ) ×G∞ A → M denote the bundle associated to J ∞ (M ) by the representation of G∞ on A and Γ(A) be its space of smooth sections on which VM acts ∞ via γ ∞ . We identify Γ(A) with C ∞ (J ∞ (M ), A)G . Then the map ? (C.7) γA : = (γ ∞ )? ? F ? : C ? (L1 , A)GLN (R) = C ? (L0 , gl (R), A) → C ? (VM , Γ(A))

c c N c

is a morphism of chain complexes. Di?erent choices of a?ne connections on M yield the same maps ? ? Hc (L1 , A)GLN (R) ? Hc (L0 , glN (R), A) → Hc (VM , Γ(A)) = ? in cohomology. Proof. For the continuity of the so obtained cochains γA (ω) of VM , we note that the continuous representation of G∞ on the ?nite-dimensional space A factors through some ?nitedimensional quotient group Gk . Hence the bundle A is also associated to J k (M ), Γ(A) can be realized as A-valued Gk -equivariant functions on J k (M ), and the action of VM on this space via γ k : VM → VJ k (M) is continuous because the ?nite dimensionality of J k (M ) implies that γ k is a continuous morphism of topological Lie algebras. To see that the choice of connection has no e?ect on the corresponding map in cohomology, let ?′ be another a?ne connection on M and observe that ?t := t?′ + (1 ? t)? de?nes a smooth family of a?ne connections on M with ?0 = ? and ?1 = ?′ . It is easy to see that the corresponding functions s1 : J 1 (M ) → J ∞ (M ) depend smoothly on t, and so do the p corresponding G1 -equivariant functions Ft : J ∞ (M ) → G1 . Hence, for each ω ∈ Cc (L1 , A)GLN (R) ? ? the di?erential forms F1 ω?F0 ω are equivariantly exact, i.e., the di?erential of a G∞ -equivariant A-valued (p ? 1)-form (cf. [Ko74, p.143]). This implies the assertion. Remark C.4. We also note that the 1 -form δ(F ) ∈ ?1 (J ∞ (M ), L1 ) satis?es the Maurer– Cartan equation, hence de?nes a crossed homomorphism δ(F ): V(J ∞ (M )) → C ∞ (J ∞ (M ), L1 ) (Proposition I.8). Composing with the homomorphism γ ∞ : VM → VJ ∞ (M) , we thus obtain a crossed homomorphism ∞ δ(F ) ? γ ∞ : VM → C ∞ (J ∞ (M ), L1 )G . That δ(F ) ? γ ∞ maps into G∞ -equivariant functions is due to (C.3) and the G∞ -equivariance p of F , resulting from (C.4). We further note that any element ω ∈ Cc (L1 , A)GLN (R) de?nes ∞ ∞ ∞ by composition a VM -equivariant p-cochain of the Lie algebra C (J (M ), L1 )G with values ∞ in C ∞ (J ∞ (M ), A)G ? Γ(A). Now Theorem I.7 shows that pulling back with the crossed = homomorphism δ(F ) ? γ ∞ yields Γ(A)-valued Lie algebra cochains and that this is compatible with the mutual di?erentials.

On the cohomology of vector ?elds on parallelizable manifolds

25

Example C.5. For A = Λk (V ′ ), V = RN , and the canonical representation of G1 ? = GLN (R) ? G∞ /G1 on this space, we obtain for the space of smooth sections Γ(A) ? ?k , = M = so that we get a morphism of cochain complexes

? ? γA : Cc (L1 , Λk (V ′ ))GLN (R) → Cc (VM , ?k ). M

In this case the bundle A = Λk (T ? (M )) is associated to the frame bundle J 1 (M ) which creates a simpler picture than working with the in?nite-dimensional manifold J ∞ (M ). Example C.6. If A = R is the trivial module, we obtain in particular FM -valued cocycles on VM from any map k p (γ k )? : (?p k (M) )G → Cc (V(M ), FM ). J Here is a concrete example: For k = 1 we consider the 1 -form ω ∈ ?1 (J 1 (M ), R)GLN (R) de?ned as follows. From the homomorphism χ: GLN (R) → R× , + g → | det(g)| we obtain an associated bundle J 1 (M ) ×χ R× , and since R× is contractible, this bundle has a + + global section (we could also take log ?χ and obtain an a?ne bundle), which means that there is a smooth function F : J 1 (M ) → R× + Then δ(F ) ∈ ?1 (J 1 (M ), R)GLN (R) is a GLN (R)-invariant 1 -form. If ? is a volume form on M , then we can construct F directly from ? by F (v1 , . . . , vN ) := |?m (v1 , . . . , vN )| for any basis (v1 , . . . , vN ) on Tm (M ). For a di?eomorphism ? ∈ Di?(M ), we then have (?.F )(v1 , . . . , vN ) = F (T (??1 ).(v1 , . . . , vN )) = |?(T (??1 ).(v1 , . . . , vN ))| = |(??1 )? ?(v1 , . . . , vN )|. Dividing by F (v1 , . . . , vN ), we obtain a smooth function (?.F )F ?1 ∈ C ∞ (M, R× ), + and passing to the Lie derivative, we obtain for each vector ?eld X on M a smooth function (LX .F )F ?1 ∈ C ∞ (M, R). Now we turn to Tsujishita’s construction of the homomorphism

? ? Hc (L1 , Λk (V ′ ))GL(V ) → Hc (VM , ?k ) M

with

F (α.g) = F (α)| det(g)|

for

g ∈ GLN (R), α ∈ J 1 (M ).

in term of the cocycles LX ? associated to a?ne connections (Lemma II.1). Let ? be an a?ne connection on M . As in Lemma C.1, we obtain from ? a smooth section 2 2 s: J 1 (M ) → J 2 (M ), s(α) := j0 (Exp? ?α) = j0 (Exp? ) ? α. m m From s(αg) = s(α)g for β ∈ GL(V ) it follows that s is GL(V )-equivariant. Let F : J 2 (M ) → G1 /G2 ? Sym2 (V, V ) denote the unique smooth G1 -equivariant smooth = function vanishing on s(J 1 (M )). Identifying Sym2 (V, V ) ? L1 /L2 with the corresponding = subspace of V ′ ? gl(V ) ? (V ′ ? V ′ ) ? V corresponds to composition with the map ev0 dJ: L1 → = V ′ ? gl(V ). The map 2 δ(F ) ? γ 2 : VM → C ∞ (J 2 (M ), L1 /L2 )G is a crossed homomorphism (Proposition I.8), and since L1 /L2 is abelian, it is a 1 -cocycle. Composing with ev0 dJ: L1 /L2 → V ′ ? gl(V ), we thus get a 1 -cocycle ev0 dJδ(F ) ? γ 2 : VM → C ∞ (J 2 (M ), V ′ ? gl(V ))G ? C ∞ (J 1 (M ), V ′ ? gl(V ))GL(V ) ? ?1 (M, End(T M )). = = The following theorem is the link between the approaches in [Tsu81] and [Ko74].

2

26 Theorem C.7.

Yuly Billig, Karl-Hermann Neeb

If ? is torsion free, then ev0 dJδ(F )(γ 2 (X)) = LX ? ∈ ?1 (M, End(T M )).

Proof. Since this is a local assertion, it su?ces to assume that M is an open subset of V ? RN . Then we write = ?X Y = dY (X) + Γ(X, Y ) for a symmetric (1, 2)-tensor Γ (recall that Γ is symmetric if and only if ? is torsion free). We then identify J 1 (M ) = M × GL(V ) and J 2 (M ) = M × G2 ? M × (Sym2 (V, V ) ? GL(V )). =

Here we write elements of G2 as α = α1 + α2 with α1 ∈ GL(V ) linear and α2 quadratic. Then the group structure of G2 is given by α ? β = α1 β1 + (α1 β2 + α2 β1 ) = α1 β + α2 β1 = αβ1 + α1 β2 . From this we see that (C.10) α?1 = α?1 ? α?1 α2 α?1 . 1 1 1

To describes the section s: J 1 (M ) → J 2 (M ) explicitly, let γv (t) = Exp? (tv) be the geodesic ′ with γv (0) = m and γv (0) = v (which is de?ned for t su?ciently close to 0 ). Then the relation

′′ ′ ′ γv (t) = ?Γ(γv (t), γv (t)) ′′ leads to γv (0) = ?Γ(v, v), so that 2 1 1 j0 (Exp? )(v) = v + 2 γ ′′ (0) = v ? 2 Γ(v, v). m

We thus obtain

1 s(m, α1 ) = (m, (1 ? 2 Γ)α1 ).

Accordingly, we ?nd

1 1 F (m, α) = α?1 (α + 2 Γ ? α1 ) = 1 + α?1 α2 + 2 α?1 Γ ? α1 1 1 1

because F (m, α(1 + β2 )) = F (m, α + α1 β2 ) = F (m, α) + β2 = F (m, α).β2 and

?1 F (m, αβ1 ) = β1 F (m, α)β1 .

From the relation

?1 ?1 ?1 (β ? α)1 (β ? α)2 = α?1 β1 (β1 α2 + β2 α1 ) = α?1 α2 + α?1 β1 β2 α1 , 1 1 1

we obtain the following formula for the di?erential of F in (v, β ? α) ∈ T(m,α) (M × G2 ) with β ∈ T1 (G2 ) ? L0 /L2 : =

1 dF (m, α)(v, β ? α) = 2 α?1 dΓ(v) ? α1 + α?1 β2 α1 ? 1 α?1 (β1 .Γ) ? α1 1 1 2 1

= α?1 1

1 2 dΓ(v)

1 + β2 ? 2 (β1 .Γ) ? α1 .

The natural lift of X ∈ VM to J 2 (M ) is given by

2 γ 2 (X)(m, α) = j0 (X ? α) = (X(m), J(X)m ? α + 1 (d2 X)m ? α1 ) 2 1 = X(m), J(X)m + 2 (d2 X)m ? α

On the cohomology of vector ?elds on parallelizable manifolds

27

(J(X)m v = (dX)m (v) denoting the Jacobian of X ), so that

1 γ 2 (X)(s(m, α1 )) = X(m), J(X)m ? α1 ? 2 J(X)m Γ ? α1 + 1 (d2 X)m ? α1 . 2

This leads to dF (γ 2 (X))(s(m, α1 )) = δ(F )(γ 2 (X))(s(m, α1 ))

1 = 2 α?1 dΓ(X(m)) + (d2 X)m ? J(X)m .Γ ? α1 . 1

This is a GL(V )-equivariant function J 1 (M ) → Sym2 (V, V ), so that the corresponding (1, 2)tensor ?eld is given in the canonical coordinates by the smooth function M → Sym2 (V, V ) given by m → dF (γ 2 (X))(s(m, 1)) = 1 dΓ(X(m)) + (d2 X)m ? J(X)m Γ . 2 Suppose that β: V → V is a quadratic map and β the corresponding symmetric bilinear map determined by β(v) = 1 β(v, v). Considering β as an element of L0 /L2 , we have 2 J(β)(v) = (dβ)v = β(v, ·) ∈ gl(V ) and thus (dJ(β))0 (v)v = ev0 (dJ(β))(v)(v) = β(v, v) = 2β(v). Applying ev0 ?dJ to the smooth function above thus leads to ev0 dJδ(F )(γ 2 (X))(s(m, 1)) = dΓ(X(m)) + (d2 X)m ? J(X)m Γ ∈ V ′ ? gl(V ). In view of Remark II.2, it remains to show that LX Γ = dΓ(X) ? J(X).Γ. It su?ces to verify this relation by evaluating on constant vector ?elds Y and Z ∈ V : (LX Γ)(Y, Z) = [X, Γ(Y, Z)] ? Γ([X, Y ], Z) ? Γ(Y, [X, Z]) = (dΓ(X))(Y, Z) ? dX(Γ(Y, Z)) + Γ(dX(Y ), Z) + Γ(Y, dX(Z)) = (dΓ(X))(Y, Z) ? (J(X).Γ)(Y, Z).

Now we can identify the image of the Λk (V ′ )-valued cocycle ΨL (Theorem B.2) unk der Tsujishita’s map γΛk (V ′ ) , described in Proposition C.3, which maps it into an element of k Zc (VM , ?k ). M Theorem C.8. For each torsion free connection ? on the manifold M , we have γΛk (V ′ ) (ΨL ) = Ψk . k Proof. We ?rst observe that ΨL = (ev0 dJ)? fk , k is given by ((α1 ? A1 ), . . . , (αk ? Ak )) →

σ∈Sk

where

fk : (V ′ ? End(V ))k → Λk (V ′ )

sgn(σ) Tr(Aσ(1) · · · Aσ(k) )ασ(1) ∧ · · · ∧ ασ(k) Tr(Aσ(1) · · · Aσ(k) ) · α1 ∧ · · · ∧ αk .

σ∈Sk

=

28

Yuly Billig, Karl-Hermann Neeb

In this sense, we have fk = mk ?βk , where mk : (V ′ )?k → Λk (V ′ ) is the alternating multiplication map and βk : End(V )k → R is the symmetric k -linear map de?ned by βk (A1 , . . . , Ak ) =

σ∈Sk

Tr(Aσ(1) · · · Aσ(k) ).

According to Proposition C.3, we have for X1 , . . . , Xk ∈ VM : γΛk (V ′ ) (ΨL )(X1 , . . . , Xk ) = (F ? ΨL )(γ 2 (X1 ), . . . , γ 2 (Xk )), k k where we identify

2 ∞ ?k ? C ∞ (J ∞ (M ), Λk (V ′ ))G ? C ∞ (J 2 (M ), Λk (V ′ ))G ? C ∞ (J 1 (M ), Λk (V ′ ))GL(V ) . = = M =

This identi?cation is based on the canonical map J ∞ (M ) × Λk (V ′ ) → Λk (T ? (M )), ([α], ω) → ω ? α?1 1

which is constant on the diagonal G∞ -action and factors through the corresponding map on J 1 (M ), resp., J 2 (M ). Since the cocycles ΨL factor through the quotient algebra L0 /L2 , the map γΛk (V ′ ) can be k constructed with J 2 (M ) instead of J ∞ (M ) and to pass from G2 -equivariant smooth functions on J 2 (M ) to GL(V )-equivariant smooth functions on J 1 (M ), we may simply restrict to s(J 1 (M )), the 1-level set of the function F . We thus obtain with Theorem C.7 for any torsion free connection ?: γΛk (V ′ ) (ΨL )(X1 , . . . , Xk ) = (F ? ΨL )(γ 2 (X1 ), . . . , γ 2 (Xk )) k k = (?1)k ΨL (δ(F )(γ 2 (X1 )), . . . , δ(F )(γ 2 (Xk ))) k = (?1)k fk ev0 dJδ(F )(γ 2 (X1 )), . . . , ev0 dJδ(F )(γ 2 (Xk )) = (?1)k fk (LX1 ?, . . . , LXk ?) = Ψk (X1 , . . . , Xk ), where we identify LX ? with an element of ?1 (M, End(T M )) ? C ∞ (J 1 (M ), V ′ ? End(V ))GL(V ) = and use ?k ? C ∞ (J 1 (M ), Λk (V ′ ))GL(V ) . = M Proof of Theorem III.1: We choose an a?ne torsion free connection ?. From the formulation of Tsujishita’s Theorem in [Tsu81], Thm. 5.1.6, we get (with V = RN ) an isomorphism of bigraded algebras:

? ? ? Hc (VM , ?? ) ? Hc (VM , FM ) ? Hc (L0 , glN (R), Λ? (V ′ )). M = ? We further know from Theorem B.2, that Hc (L0 , glN (R), Λ? (V ′ )) is generated by the classes of L the cocycles Ψk . Hence it su?ces to observe with Theorem C.8 that Tsujishita’s map ? ? γΛk (V ′ ) : Cc (L1 , Λk (V ′ ))GLN (R) ? Cc (L0 , glN (R), Λk (V ′ )) → Cc (VM , ?k ) = ? M

(Proposition C.3) maps ΨL to Ψk . k

On the cohomology of vector ?elds on parallelizable manifolds

29

Appendix D. Cohomology of vector ?elds with values in the trivial module

Here we shall review the results of Hae?iger [Hae76]. We begin by recalling the de?nition of the Weil algebra WN . Let E(u1 , . . . , uN ) be the exterior algebra in the generators u1 , . . . , uN , and let R[c1 , . . . , cN ], be the polynomial algebra in the generators c1 , . . . , cN . Introduce gradings on these algebras by assigning degrees to the generators: deg(uk ) = 2k ? 1 and deg(ck ) = 2k. Consider the quotient R[c1 , . . . , cN ] of the polynomial algebra R[c1 , . . . , cN ] by the ideal spanned by the elements of degrees exceeding 2N . The Weil algebra is the di?erential graded algebra WN = E(u1 , . . . , uN ) ? R[c1 , . . . , cN ] with the di?erential de?ned by d(uk ) = ck and d(ck ) = 0.

Let H ? (WN ) be the cohomology of this di?erential graded algebra. Gelfand–Fuks used the Weil algebra to describe the cohomology of the Lie algebra WN of formal vector ?elds in dimension N :

? Theorem D.1. ([GF70b]) Hc (WN , R) ? H ? (WN ) = The explicit description of the cohomology of the Weil algebra was given by Vey:

Theorem D.2. (D.1)

([Go74]) H ? (WN ) is spanned by 1 and the cocycles (ui1 ∧ . . . ∧ uir ) ? (cj1 . . . cjs )

satisfying the following conditions: 1 ≤ i1 < . . . < ir ≤ N ; 1 ≤ j1 ≤ . . . ≤ js ≤ N ; i1 ≤ j1 ; r > 0 ; j1 + . . . + js ≤ N ; i1 + j1 + . . . + js > N . Combining the preceding two theorems, we derive: Corollary D.3. (a) H q (WN , R) = 0 for 1 ≤ q ≤ 2N and for q ≥ (N + 1)2 . (b) dim H 2N +1 (WN , R) = p(N + 1) ? 1 , where p is the partition function. (c) dim H 2N +2 (WN , R) = 0 . Proof. (a) The degree of the generator (D.1) is at least 2i1 ? 1 + 2(j1 + . . . + js ) > 2N ? 1, and since it is odd, it is at least 2N + 1 . On the other hand, it is at most 1 + 3 + . . . + (2N ? 1) + 2(j1 + . . . + js ) ≤ N 2 + 2N < (N + 1)2 . (b) If the degree is exactly q = 2N + 1 , then the argument in (a) implies 2i1 ? 1 + 2(j1 + . . . + js ) = 2N + 1, which leads to r = 1 and i1 + j1 + . . . + js = N + 1 . Only the partition N + 1 = (N + 1) + 0 + . . . + 0 does not contribute, and this proves (b).

30

Yuly Billig, Karl-Hermann Neeb

(c) If the degree q of the generator in (D.1) is even and > N , then j1 + . . . + js ≤ N implies r ≥ 2 and hence q = (2i1 ? 1) + (2i2 ? 1) + 2(j1 + . . . + js ) > 2N ? 1 + 3 = 2N + 2. Let W (N ) be the graded vector space

(N +1)2 ?2

W (N ) =

q=2N

W (N )q ,

where dim W (N )q = dim H q+1 (WN ). We view W (N ) as a super vector space, where the parity of W (N )q is equal to the parity of q . Let LN be the free Lie superalgebra generated by W (N ) (see e.g., [BMPZ92] for the description of the basis of a free Lie superalgebra). The Z-grading on W (N ) extends to a Z-grading of LN . This Lie superalgebra was introduced by Hae?iger [Hae76] in order to give the description of the cohomology of the Lie algebra of smooth vector ?elds on a manifold with coe?cients in the trivial module. Let V be a graded vector space obtained from LN by a shift in the grading: Vq = (LN )q?1 . Since the grading of LN starts at component 2N , the grading of V starts at component 2N + 1 . Let ?? be the di?erential graded algebra of smooth di?erential forms on M , and consider M the tensor product space ?? ? V with the grading M deg(ω ? v) = deg v ? deg ω, ω ∈ ?? , v ∈ V . M

Since ?p vanishes for p > N , the grading on the space ?? ? V starts at degree N + 1 . Next M M we consider the graded algebra S ? (?? ? V ) of supersymmetric multilinear forms on the graded M superspace ?? ? V . We have M S ? (?? ? V ) = ⊕ S p (?? ? V ), M M

p=0 ∞

where S p (?? ? V )q = 0 for q < p · (N + 1). Hence, apart from the component of degree 0, the M grading on S ? (?? ? V ) starts in degree N + 1 . M Theorem D.4. ([Hae76], Theorem A) Let M be an N -dimensional smooth manifold. On the graded algebra S ? (?? ? V ) there exists a di?erential of degree 1, which depends on a choice of M the representatives of the Pontrjagin classes in ?? , and a homomorphism of di?erential graded M algebras ? S ? (?? ? V ) → Cc (VM , R), M which induces an isomorphism in cohomology. Corollary D.5.

s Hc (VM , R) = 0 for 1 ≤ s ≤ N .

If all Pontrjagin classes of M vanish and there is a splitting algebra homomorphism ? HM ?→ ?? (which is, for example, the case when M is a torus or a compact Lie group), M Hae?iger gives a more explicit realization for the cohomology of VM , which we described now. ? ? Let HM be the cohomology algebra of the manifold M . We view HM as a commutative superalgebra and form the graded Lie superalgebra L(M ) = H ? (M, R) ? LN with the bracket of homogeneous elements given by [ω ? x, ω ′ ? x′ ] = (?1)deg x·deg ω ωω ′ ? [x, x′ ],

′

On the cohomology of vector ?elds on parallelizable manifolds

31

and the grading by L(M )r =

p?q=r p HM ? (LN )q .

It follows in particular that L(M )q = 0 for q < N . Consider the cohomology H ? (L(M )) of the Lie superalgebra L(M ) (see [Fu86], Sect. 1.6.3 for the de?nition of the cohomology for the Lie superalgebras, cf. Sect. 1.2 in [Hae76]). Since L(M ) is graded, its cohomology inherits a grading H p (L(M ), R) = ⊕q H p (L(M ), R)q (see [Fu86], Sect. 1.3.7). Since the grading of L(M ) starts at degree N , we get that (D.2) H p (L(M ), R)q = 0 for q < p · N.

Theorem D.6. ([Hae76], Theorem 3.4) Let M be a smooth manifold with a ?nite-dimensional ? cohomology HM for which all Pontryagin classes vanish. Suppose, moreover, that there exists ? a homomorphism of di?erential graded algebras HM → ?? , which induces an isomorphism in M cohomology (this is the case if M is a compact Lie group). Then

s Hc (VM , R) = p+q=s

⊕ H p (L(M ), R)q .

Corollary D.7. Under the assumptions of the preceding theorem, we have s (a) Hc (VM , R) = 0 for 1 ≤ s ≤ N .

s (b) For N + 1 ≤ s ≤ 2N + 1 , dim Hc (VM , R) = k dim HM · dim W (N )s+k?1 . k=0 N N ? (c) dim Hc +1 (VM , R) = p(N +1)?1 and only HM = R contributes in the decomposition above. N ?1 1 N (d) If b1 (M ) := dim HM = dim HM , then dim Hc +2 (VM , R) = b1 (M ) · (p(N + 1) ? 1) and N ?1 ? b1 (M) only HM = R contributes in the decomposition. Proof. (a) If H p (L(M ), R)q is non-zero, then q ≥ pN , so that (D.2) implies s = p + q ≥ N

(N + 1)p ≥ N + 1 if s = 0 . s (b) We note that (D.2) implies Hc (VM , R) ? H 1 (L(M ), R)s?1 for N + 1 ≤ s ≤ 2N + 1 . = However,

? H 1 (L(M ), R) = (L(M )/[L(M ), L(M )]) = HM ? W (N ) ? ?

and the claim (b) follows. If M is an N -dimensional torus TN , its cohomology H ? (TN ) is an exterior algebra in N generators of degree 1. Combining the two previous corollaries, we get with b1 (TN ) = N : Corollary D.8. For the torus T = TN , we have: s (a) Hc (VT , R) = 0 for 1 ≤ s ≤ N . N (b) dim Hc +1 (VT , R) = p(N + 1) ? 1 . N +2 (c) dim Hc (VT , R) = N · (p(N + 1) ? 1). Remark D.9. For M = Sn , the algebra H ? (VSn , R) is calculated by Cohen and Taylor in [CT78]; Theorems 1.3 and 3.2.

References [AMR83] [ABFP05] [BMPZ92] Abraham, R., J. E. Marsden, and T. Ratiu, “Manifolds, Tensor Analysis, and Applications,” Addison-Wesley, 1983. Allison, B., S. Berman, J. Faulkner, and A. Pianzola, Realizations of gradedsimple algebras as loop algebras. math.RA/0511723.. Bahturin, Y. A., Mikhalev, A. A., Petrogradsky, V. M., Zaicev, M. V., “In?nitedimensional Lie superalgebras,” Walter de Gruyter & Co., Berlin, 1992.

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[BN06] [Bi06] [ChE48] [CT78]

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[EM94]

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On the cohomology of vector ?elds on parallelizable manifolds

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[Ne04] [Ne06a] [Ne06b] [Neh04] [PS86] [Ro71] [Tsu77] [Tsu81]

Maier, P., Central extensions of topological current algebras, in “Geometry and Analysis on Finite- and In?nite-Dimensional Lie Groups,” A. Strasburger et al Eds., Banach Center Publications 55, Warszawa 2002; 61–76. Neeb, K.-H., Abelian extensions of in?nite-dimensional Lie groups, Travaux math?matiques 15 (2004), 69–194. e —, Lie algebra extensions and higher order cocycles, J. Geom. Sym. Phys. 5 (2006), 48–74. —, Non-abelian extensions of topological Lie algebras, Communications in Algebra 34 (2006), 991–1041. Neher, E., Extended a?ne Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26:3 (2004), 90–96. Pressley, A., and G. Segal, “Loop Groups,” Oxford University Press, Oxford, 1986. Rosenfeld, B. I., Cohomology of certain in?nite-dimensional Lie algebras, Funct. Anal. Appl. 13 (1971), 340–342. Tsujishita, T., On the continuous cohomology of the Lie algebra of vector ?elds, Proc. Jap. Math. Soc. 53:A (1977), 134–138. —, “Continuous cohomology of the Lie algebra of vector ?elds,” Memoirs of the Amer. Math. Soc. 34:253, 1981.

Yuly Billig School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 Canada billig@math.carleton.ca

Karl-Hermann Neeb Technische Universit¨t Darmstadt a Schlossgartenstrasse 7 D-64289 Darmstadt Deutschland neeb@mathematik.tu-darmstadt.de

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Semisimple Frobenius (super)*manifolds* and quantum *cohomology* *of* $P^r$ We ...at connection *on* a *vector* bundle *on* P1 having two singular points, a ...

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