On the cohomology of vector fields on parallelizable manifolds
[Sur la cohomologie des champs vectoriels sur les variétés parallélisables]
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 Ω ¯ M p =Ω M p /dΩ 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 Ω ¯ M 1 , généralisant les algèbres de Kac-Moody affines. L’espace H 2 (𝒱 M ,Ω ¯ 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 Ω ¯ M p =Ω M p /dΩ 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 Ω ¯ M 1 , generalizing affine Kac-Moody algebras. The second cohomology H 2 (𝒱 M ,Ω ¯ M 1 ) classifies twists of the semidirect product of 𝒱 M with the universal central extension of a gauge Lie algebra.

Reçu le :
Accepté le :
DOI : https://doi.org/10.5802/aif.2402
Classification : 17B56,  17B65,  17B68
Mots clés : algèbre de Lie des champs vectoriels, cohomologie de l’algèbre de Lie, cohomologie de Gelfand-Fuks, algèbre de Lie affine étendu
@article{AIF_2008__58_6_1937_0,
     author = {Billig, Yuly and Neeb, Karl-Hermann},
     title = {On the cohomology of vector fields on parallelizable manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {1937--1982},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     doi = {10.5802/aif.2402},
     zbl = {1157.17007},
     mrnumber = {2473625},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2402/}
}
Billig, Yuly; Neeb, Karl-Hermann. On the cohomology of vector fields on parallelizable manifolds. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1937-1982. doi : 10.5802/aif.2402. https://aif.centre-mersenne.org/articles/10.5802/aif.2402/

[1] Abraham, R.; Marsden, J. E.; Ratiu, T. Manifolds, Tensor Analysis, and Applications, Addison-Wesley, 1983 | MR 697563 | Zbl 0508.58001

[2] Allison, B.; Berman, S.; Faulkner, J.; Pianzola, A. Realizations of graded-simple algebras as loop algebras (math.RA/0511723) | Zbl pre05309524

[3] Bahturin, Y. A.; Mikhalev, A. A.; Petrogradsky, V. M.; Zaicev, M. V. Infinite-dimensional Lie superalgebras, Walter de Gruyter & Co, 1992 | MR 1192546 | Zbl 0762.17001

[4] Beggs, E. J. The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford, Tome 38 (1987) no. 2, pp. 131-154 | Article | MR 891612 | Zbl 0636.58004

[5] Benkart, G.; Neher, E. The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra, Tome 205 (2006) no. 1, pp. 117-145 | Article | MR 2193194 | Zbl pre05004396

[6] Berman, S.; Billig, Y. Irreducible representations for toroidal Lie algebras, J. Algebra, Tome 221 (1999), pp. 188-231 | Article | MR 1722910 | Zbl 0942.17016

[7] Bernshtein, I. N.; Rozenfel’d, B. I. Homogeneous spaces of infinitedimensional Lie algebras and characteristic classes of foliations, Russ. Math. Surveys, Tome 28 (1973) no. 4, pp. 107-142 | Article | Zbl 0289.57011

[8] Billig, Y. A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not., 2006 (Art. ID 68395, 46 pp.) | MR 2272091 | Zbl pre05136974

[9] Chevalley, C.; Eilenberg, S. Cohomology theory of Lie groups and Lie algebras, Transactions of the Amer. Math. Soc., Tome 63 (1948), pp. 85-124 | Article | MR 24908 | Zbl 0031.24803

[10] Cohen, F. R.; Taylor, L. R. 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.) Tome 657 (1978), pp. 106-173 | MR 513543 | Zbl 0398.55004

[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., Tome 62 (1983), pp. 197-214 | MR 713396 | Zbl 0481.58032

[12] Eswara Rao, S.; Moody, R. V. Vertex representations for n-toroidal Lie algebras and a generalization of the Virasoro algebra, Comm. Math. Phys., Tome 159 (1994), pp. 239-264 | Article | MR 1256988 | Zbl 0808.17018

[13] Feigin, B. L.; Fuchs, D. B. Cohomologies of Lie Groups and Lie Algebras, Lie Groups and Lie Algebras II (Encyclop. Math. Sci.) Tome 21 (2001) | Zbl 0931.17014

[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, Tome 291 (1980) no. 4, p. A331-A335 | Zbl 0462.58011

[15] Fuks, D. B. Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, New York, London, 1986 | MR 874337 | Zbl 0667.17005

[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 266195 | Zbl 0216.20302

[17] Gelfand, I. M.; Fuks, D. B. Cohomology of the Lie algebra of vector fields with nontrivial coefficients, Func. Anal. and its Appl., Tome 4 (1970), pp. 181-192 | Article | MR 287589 | Zbl 0222.58001

[18] Godbillon, C. Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki (1972/1973), Exp. No. 421 (Lecture Notes in Math.) Tome 383 (1974), pp. 69-87 | Numdam | Zbl 0296.17010

[19] Haefliger, A. Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. Ec. Norm. Sup., 4e série, Tome 9 (1976), pp. 503-532 | Numdam | Zbl 0342.57014

[20] Hochschild, G.; Serre, J.-P. Cohomology of Lie algebras, Annals of Math., Tome 57 (1953) no. 3, pp. 591-603 | Article | MR 54581 | Zbl 0053.01402

[21] Kassel, C. Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure Applied Algebra, Tome 34 (1984), pp. 265-275 | Article | MR 772062 | Zbl 0549.17009

[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, Tome 7 (1974), pp. 139-153 | Numdam | Zbl 0316.58003

[23] Larsson, T. A. Lowest-energy representations of non-centrally extended diffeomorphism algebras, Comm. Math. Phys., Tome 201 (1999), pp. 461-470 | Article | MR 1682285 | Zbl 0936.17025

[24] Maier, P. Central extensions of topological current algebras, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, Tome 55 (2002), pp. 61-76 | MR 1911980 | Zbl 1045.17008

[25] Neeb, K.-H. Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques, Tome 15 (2004), pp. 69-194 | MR 2143422 | Zbl 1079.22018

[26] Neeb, K.-H. Lie algebra extensions and higher order cocycles, J. Geom. Sym. Phys., Tome 5 (2006), pp. 48-74 | MR 2269881 | Zbl 1105.53064

[27] Neeb, K.-H. Non-abelian extensions of topological Lie algebras, Communications in Algebra, Tome 34 (2006), pp. 991-1041 | Article | MR 2208114 | Zbl pre05018918

[28] Neher, E. Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can., Tome 26 (2004) no. 3, pp. 90-96 | MR 2083842 | Zbl 1072.17012

[29] Pressley, A.; Segal, G. Loop Groups, Oxford University Press, Oxford, 1986 | MR 900587 | Zbl 0618.22011

[30] Rosenfeld, B. I. Cohomology of certain infinite-dimensional Lie algebras, Funct. Anal. Appl., Tome 13 (1971), pp. 340-342 | Zbl 0248.57030

[31] Tsujishita, T. On the continuous cohomology of the Lie algebra of vector fields, Proc. Jap. Math. Soc., Tome 53:A (1977), pp. 134-138 | MR 458517 | Zbl 0476.58032

[32] Tsujishita, T. Continuous cohomology of the Lie algebra of vector fields, Memoirs of the Amer. Math. Soc., Tome 253 (1981) no. 34, 154p. pages | MR 634471 | Zbl 0482.58036