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On the cohomology of vector fields on parallelizable manifolds. (English) Zbl 1157.17007
Let \({\mathcal V}_M\) be the Lie algebra of smooth vector fields on a parallelizable smooth manifold \(M\) of dimension \(N\). \({\mathcal V}_M\) is a Fréchet-Lie algebra. It acts continuously on the space of \(p\)-forms \(\Omega^p_M\) on \(M\), and on its factor space \(\overline{\Omega}^p_M\,=\,\Omega^p_M\,/\,d\Omega^{p-1}_M\) by the space of exact \(p\)-forms \(d\Omega^{p-1}_M\) by Lie derivative. The goal of the article under review is the computation of the Gelfand-Fuks cohomology spaces \(H^2_c({\mathcal V}_M,\overline{\Omega}^p_M)\). The index \(c\) in this formula stands for continuous cohomology, i.e. all cochains are considered to be continuous with respect to the fixed Fréchet topologies on Lie algebra and module. Motivation for this work comes from the case of the circle \(M=S^1\) and \(p=1\), where \(\overline{\Omega}^1_{S^1}\) is the center of the universal central extension of the current Lie algebra \({\mathcal C}^{\infty}(S^1)\otimes{\mathfrak g}\), \({\mathfrak g}\) being a finite dimensional complex simple Lie algebra.
The first part of the article explains the construction of the cocycles which are going to be (representatives of) the generators of the above cohomology spaces. Namely, given two Lie algebras \({\mathfrak h}\) and \({\mathfrak n}\), Billig and Neeb show in the beginning how a crossed homomorphism \(\theta:{\mathfrak h}\to{\mathfrak n}\) induces a morphism of complexes \[ C^*_{\text{eq}}({\mathfrak n},V)\to C^*({\mathfrak h},V), \] from the complex of equivariant cochains on \({\mathfrak n}\) to cochains on \({\mathfrak h}\), \(V\) being an \({\mathfrak h}\)-module, seen as a trivial \({\mathfrak n}\)-module. The next step is to use Koszul’s construction of \(1\)-cocycles \(\zeta:=(X\mapsto {\mathcal L}_X\nabla)\) on \({\mathcal V}_M\) with values in \(\Omega^1(M,\text{End}(TM))\) for a given affine connection \(\nabla\). In case the connection \(\nabla\) is given by a trivialization of \(TM\) via the Maurer-Cartan form \(\kappa\in\Omega^1(M,{\mathbb R}^N)\), the relation \({\mathcal L}_X\kappa=-\theta(X)\kappa\) holds for some \(\theta\in{\mathcal C}^{\infty}(M,{\mathfrak g}{\mathfrak l}({\mathbb R}^N))\), which happens to be a crossed homomorphism. Therefore the above results apply and this serves to construct cocycles \[ \Psi_k(X_1,\ldots,X_k)\,=\,\sum_{\sigma\in S_k}\text{sgn}(\sigma)\text{Tr} (d\theta(X_{\sigma(1)})\wedge\ldots\wedge d\theta(X_{\sigma(k)})) \] and \[ \overline{\Psi}_k(X_1,\ldots,X_k)=\sum_{\sigma\in S_k}\text{sgn}(\sigma) \text{Tr}(\theta(X_{\sigma(1)})\wedge d\theta(X_{\sigma(2)})\wedge\ldots\wedge d\theta(X_{\sigma(k)})). \]

The second part of the article joins now these natural generators to T. Tsujishita’s computation [“Continuous cohomology of the Lie algebra of vector fields.” Mem. Am. Math. Soc. 253 (1981; Zbl 0482.58036)] of cohomology spaces of the type \(H^*_c({\mathcal V}_M,A_M)\), the cohomology with values in the tensor spaces \(A_M\) starting from the cohomology with values in the functions \({\mathcal C}^{\infty}(M)\). For the proof of their main result (theorem 3.1), Billig and Neeb go through large parts of Tsujishita’s work to show that the above method for constructing cocycles supplies generators of \(H^*_c({\mathcal V}_M,\Omega^p_M)\).
Finally, the authors use short exact sequences to pass from \(\Omega^p_M\) to \(\overline{\Omega}^p_M\). In the end, the result (theorem 4.5) reads: let \(n\geq 2\), then there is an isomorphism \[ H^2_c({\mathcal V}_M,\overline{\Omega}^1_M)\,\cong H^3_M\oplus{\mathbb R} [\overline{\Psi}_1\wedge\Psi_1]\oplus{\mathbb R}[\overline{\Psi}_2], \] \(H^*_M\) being the de Rham cohomology of \(M\). The article is clearly written and easily accessible.

MSC:
17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
17B68 Virasoro and related algebras
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
58H10 Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
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