On the cohomology of vector fields on parallelizable manifolds

Billig, Yuly; Neeb, Karl-Hermann

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1937-1982.

Dans le présent article, nous déterminons, pour chaque variété parallélisable compacte lisse $M$, les espaces de seconde cohomologie de l’algèbre de Lie ${𝒱}_M$ des champs vectoriels lisses sur $M$ à valeurs dans le module $\overline{\Omega }{}_M^p={\Omega }_M^p/d\Omega {}_M^{p-1}$. Le cas $p=1$ est d’un intérêt particulier puisque l’algèbre de jauge des fonctions sur $M$ à valeurs dans une algèbre de Lie simple de dimension finie possède l’extension centrale universelle avec le centre ${\overline{\Omega }}_M^1$, généralisant les algèbres de Kac-Moody affines. L’espace $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ classifie des torsions du produit semi-direct de ${𝒱}_M$ avec l’extension centrale universelle d’une algèbre de Lie de jauge.

In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra ${𝒱}_M$ of smooth vector fields on $M$ with values in the module $\overline{\Omega }{}_M^p={\Omega }_M^p/d{\Omega }_M^{p-1}$. The case of $p=1$ is of particular interest since the gauge algebra of functions on $M$ with values in a finite-dimensional simple Lie algebra has the universal central extension with center ${\overline{\Omega }}_M^1$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ classifies twists of the semidirect product of ${𝒱}_M$ with the universal central extension of a gauge Lie algebra.

[1] Abraham, R.; Marsden, J. E.; Ratiu, T. Manifolds, Tensor Analysis, and Applications, Addison-Wesley, 1983 | MR | Zbl

[2] Allison, B.; Berman, S.; Faulkner, J.; Pianzola, A. Realizations of graded-simple algebras as loop algebras (math.RA/0511723)

[3] Bahturin, Y. A.; Mikhalev, A. A.; Petrogradsky, V. M.; Zaicev, M. V. Infinite-dimensional Lie superalgebras, Walter de Gruyter & Co, 1992 | MR | Zbl

[4] Beggs, E. J. The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford, Volume 38 (1987) no. 2, pp. 131-154 | DOI | MR | Zbl

[5] Benkart, G.; Neher, E. The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra, Volume 205 (2006) no. 1, pp. 117-145 | DOI | MR

[6] Berman, S.; Billig, Y. Irreducible representations for toroidal Lie algebras, J. Algebra, Volume 221 (1999), pp. 188-231 | DOI | MR | Zbl

[7] Bernshtein, I. N.; Rozenfel’d, B. I. Homogeneous spaces of infinitedimensional Lie algebras and characteristic classes of foliations, Russ. Math. Surveys, Volume 28 (1973) no. 4, pp. 107-142 | DOI | Zbl

[8] Billig, Y. A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not., 2006 (Art. ID 68395, 46 pp.) | MR

[9] Chevalley, C.; Eilenberg, S. Cohomology theory of Lie groups and Lie algebras, Transactions of the Amer. Math. Soc., Volume 63 (1948), pp. 85-124 | DOI | MR | Zbl

[10] 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 (Lectures Notes Math.), Volume 657, Proc. Conf. Evanston, Ill (1978), pp. 106-173 | MR | Zbl

[11] 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., Volume 62 (1983), pp. 197-214 | MR | Zbl

[12] Eswara Rao, S.; Moody, R. V. Vertex representations for $n$-toroidal Lie algebras and a generalization of the Virasoro algebra, Comm. Math. Phys., Volume 159 (1994), pp. 239-264 | DOI | MR | Zbl

[13] 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 (Encyclop. Math. Sci.), Volume 21, Springer-Verlag (2001) | Zbl

[14] 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, Volume 291 (1980) no. 4, p. A331-A335 | Zbl

[15] Fuks, D. B. Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, New York, London, 1986 | MR | Zbl

[16] Gelfand, I. M.; Fuks, D. B. Cohomology of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR (1970) no. 34, pp. 322-337 | MR | Zbl

[17] Gelfand, I. M.; Fuks, D. B. Cohomology of the Lie algebra of vector fields with nontrivial coefficients, Func. Anal. and its Appl., Volume 4 (1970), pp. 181-192 | DOI | MR | Zbl

[18] Godbillon, C. Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki (1972/1973), Exp. No. 421 (Lecture Notes in Math.), Volume 383 (1974), pp. 69-87 | Numdam | Zbl

[19] Haefliger, A. Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. Ec. Norm. Sup., 4e série, Volume 9 (1976), pp. 503-532 | Numdam | Zbl

[20] Hochschild, G.; Serre, J.-P. Cohomology of Lie algebras, Annals of Math., Volume 57 (1953) no. 3, pp. 591-603 | DOI | MR | Zbl

[21] Kassel, C. Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure Applied Algebra, Volume 34 (1984), pp. 265-275 | DOI | MR | Zbl

[22] 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, Volume 7, Ann. Sci. École Norm. Sup. (4) (1974), pp. 139-153 | Numdam | Zbl

[23] Larsson, T. A. Lowest-energy representations of non-centrally extended diffeomorphism algebras, Comm. Math. Phys., Volume 201 (1999), pp. 461-470 | DOI | MR | Zbl

[24] Maier, P.; Strasburger et al., A. Central extensions of topological current algebras, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, Volume 55, Banach Center Publications, Warszawa (2002), pp. 61-76 | MR | Zbl

[25] Neeb, K.-H. Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques, Volume 15 (2004), pp. 69-194 | MR | Zbl

[26] Neeb, K.-H. Lie algebra extensions and higher order cocycles, J. Geom. Sym. Phys., Volume 5 (2006), pp. 48-74 | MR | Zbl

[27] Neeb, K.-H. Non-abelian extensions of topological Lie algebras, Communications in Algebra, Volume 34 (2006), pp. 991-1041 | DOI | MR

[28] Neher, E. Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can., Volume 26 (2004) no. 3, pp. 90-96 | MR | Zbl

[29] Pressley, A.; Segal, G. Loop Groups, Oxford University Press, Oxford, 1986 | MR | Zbl

[30] Rosenfeld, B. I. Cohomology of certain infinite-dimensional Lie algebras, Funct. Anal. Appl., Volume 13 (1971), pp. 340-342 | Zbl

[31] Tsujishita, T. On the continuous cohomology of the Lie algebra of vector fields, Proc. Jap. Math. Soc., Volume 53:A (1977), pp. 134-138 | MR | Zbl

[32] Tsujishita, T. Continuous cohomology of the Lie algebra of vector fields, Memoirs of the Amer. Math. Soc., Volume 253 (1981) no. 34, pp. 154p. | MR | Zbl