On the cohomology of vector fields on parallelizable manifolds

Yuly Billig[1]; Karl-Hermann Neeb[2]

  • [1] Carleton University School of Mathematics and Statistics 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 (Canada)
  • [2] Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt (Deutschland)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 6, page 1937-1982
  • ISSN: 0373-0956

Abstract

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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 / d Ω 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.

How to cite

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Billig, Yuly, and Neeb, Karl-Hermann. "On the cohomology of vector fields on parallelizable manifolds." Annales de l’institut Fourier 58.6 (2008): 1937-1982. <http://eudml.org/doc/10366>.

@article{Billig2008,
abstract = {In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra $\{\mathcal\{V\}\}_M$ of smooth vector fields on $M$ with values in the module $\overline\{\Omega \}\,^p_M = \Omega ^p_M/d\Omega ^\{p-1\}_M$. 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 $\overline\{\Omega \}^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(\{\mathcal\{V\}\}_M, \overline\{\Omega \}^1_M)$ classifies twists of the semidirect product of $\{\mathcal\{V\}\}_M$ with the universal central extension of a gauge Lie algebra.},
affiliation = {Carleton University School of Mathematics and Statistics 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 (Canada); Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt (Deutschland)},
author = {Billig, Yuly, Neeb, Karl-Hermann},
journal = {Annales de l’institut Fourier},
keywords = {Lie algebra of vector fields; Lie algebra cohomology; Gelfand-Fuks cohomology; extended affine Lie algebra; continuous cohomology; cohomology of vector fields with values in differential forms},
language = {eng},
number = {6},
pages = {1937-1982},
publisher = {Association des Annales de l’institut Fourier},
title = {On the cohomology of vector fields on parallelizable manifolds},
url = {http://eudml.org/doc/10366},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Billig, Yuly
AU - Neeb, Karl-Hermann
TI - On the cohomology of vector fields on parallelizable manifolds
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 1937
EP - 1982
AB - In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra ${\mathcal{V}}_M$ of smooth vector fields on $M$ with values in the module $\overline{\Omega }\,^p_M = \Omega ^p_M/d\Omega ^{p-1}_M$. 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 $\overline{\Omega }^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({\mathcal{V}}_M, \overline{\Omega }^1_M)$ classifies twists of the semidirect product of ${\mathcal{V}}_M$ with the universal central extension of a gauge Lie algebra.
LA - eng
KW - Lie algebra of vector fields; Lie algebra cohomology; Gelfand-Fuks cohomology; extended affine Lie algebra; continuous cohomology; cohomology of vector fields with values in differential forms
UR - http://eudml.org/doc/10366
ER -

References

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