# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
  Abraham, R.; Marsden, J. E.; Ratiu, T., Manifolds, Tensor Analysis, and Applications, (1983), Addison-Wesley · Zbl 0508.58001  Allison, B.; Berman, S.; Faulkner, J.; Pianzola, A., Realizations of graded-simple algebras as loop algebras · Zbl 1157.17009  Bahturin, Y. A.; Mikhalev, A. A.; Petrogradsky, V. M.; Zaicev, M. V., Infinite-dimensional Lie superalgebras, (1992), Walter de Gruyter & Co · Zbl 0762.17001  Beggs, E. J., The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford, 38, 2, 131-154, (1987) · Zbl 0636.58004  Benkart, G.; Neher, E., The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra, 205, 1, 117-145, (2006) · Zbl 1163.17306  Berman, S.; Billig, Y., Irreducible representations for toroidal Lie algebras, J. Algebra, 221, 188-231, (1999) · Zbl 0942.17016  Bernshtein, I. N.; Rozenfel’d, B. I., Homogeneous spaces of infinitedimensional Lie algebras and characteristic classes of foliations, Russ. Math. Surveys, 28, 4, 107-142, (1973) · Zbl 0289.57011  Billig, Y., A category of modules for the full toroidal Lie algebra, (2006), Int. Math. Res. Not. · Zbl 1201.17016  Chevalley, C.; Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Transactions of the Amer. Math. Soc., 63, 85-124, (1948) · Zbl 0031.24803  Cohen, F. R.; Taylor, L. R.; Springer, Computations of Gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Geometric applications of homotopy theory I, 657, 106-173, (1978) · Zbl 0398.55004  de Wilde, M.; Lecomte, P. B. A., Cohomology of the Lie algebra of smooth vector fields of a manifold, associated to the Lie derivative of smooth forms, J. Math. Pures et Appl., 62, 197-214, (1983) · Zbl 0481.58032  Eswara Rao, S.; Moody, R. V., Vertex representations for $$n$$-toroidal Lie algebras and a generalization of the Virasoro algebra, Comm. Math. Phys., 159, 239-264, (1994) · Zbl 0808.17018  Feigin, B. L.; Fuchs, D. B.; Onishchik, A. L.; Vinberg, E. B., Cohomologies of Lie groups and Lie algebras, Lie Groups and Lie Algebras II, 21, (2001), Springer-Verlag · Zbl 0931.17014  Flato, M.; Lichnerowicz, A., Cohomologie des représentations définies par la dérivation de Lie et à valeurs dans LES formes, de l’algèbre de Lie des champs de vecteurs d’une variété différentiable. premiers espaces de cohomologie. applications, C. R. Acad. Sci. Paris, Sér. A-B, 291, 4, A331-A335, (1980) · Zbl 0462.58011  Fuks, D. B., Cohomology of Infinite-Dimensional Lie Algebras, (1986), Consultants Bureau, New York, London · Zbl 0667.17005  Gelfand, I. M.; Fuks, D. B., Cohomology of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR, 34, 322-337, (1970) · Zbl 0216.20302  Gelfand, I. M.; Fuks, D. B., Cohomology of the Lie algebra of vector fields with nontrivial coefficients, Func. Anal. and its Appl., 4, 181-192, (1970) · Zbl 0222.58001  Godbillon, C., Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki (1972/1973), Exp. No. 421, 383, 69-87, (1974) · Zbl 0296.17010  Haefliger, A., Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. Ec. Norm. Sup., 4e série, 9, 503-532, (1976) · Zbl 0342.57014  Hochschild, G.; Serre, J.-P., Cohomology of Lie algebras, Annals of Math., 57, 3, 591-603, (1953) · Zbl 0053.01402  Kassel, C., Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure Applied Algebra, 34, 265-275, (1984) · Zbl 0549.17009  Koszul, J.-L., Homologie des complexes de formes différentielles d’ordre supérieur, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I, 7, 139-153, (1974), Ann. Sci. École Norm. Sup. (4) · Zbl 0316.58003  Larsson, T. A., Lowest-energy representations of non-centrally extended diffeomorphism algebras, Comm. Math. Phys., 201, 461-470, (1999) · Zbl 0936.17025  Maier, P.; Strasburger et al., A., Central extensions of topological current algebras, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, 55, 61-76, (2002), Banach Center Publications, Warszawa · Zbl 1045.17008  Neeb, K.-H., Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques, 15, 69-194, (2004) · Zbl 1079.22018  Neeb, K.-H., Lie algebra extensions and higher order cocycles, J. Geom. Sym. Phys., 5, 48-74, (2006) · Zbl 1105.53064  Neeb, K.-H., Non-abelian extensions of topological Lie algebras, Communications in Algebra, 34, 991-1041, (2006) · Zbl 1158.17308  Neher, E., Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can., 26, 3, 90-96, (2004) · Zbl 1072.17012  Pressley, A.; Segal, G., Loop Groups, (1986), Oxford University Press, Oxford · Zbl 0618.22011  Rosenfeld, B. I., Cohomology of certain infinite-dimensional Lie algebras, Funct. Anal. Appl., 13, 340-342, (1971) · Zbl 0248.57030  Tsujishita, T., On the continuous cohomology of the Lie algebra of vector fields, Proc. Jap. Math. Soc., 53:A, 134-138, (1977) · Zbl 0476.58032  Tsujishita, T., Continuous cohomology of the Lie algebra of vector fields, Memoirs of the Amer. Math. Soc., 253, 34, 154p. pp., (1981) · Zbl 0482.58036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.