Lie superalgebra structures in cohomology spaces of Lie algebras with coefficients in the adjoint representation
Abstract
The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra structure. Such a description is rather imprecise: these dimensions may coincide for cohomology spaces of distinct algebras. We explicitely describe the Lie superalgebras on the space of cohomology of the maximal nilpotent subalgebra of any simple finite dimensional Lie algebra. We briefly review related results by Grozman, Penkov and Serganova, Poletaeva, and Tolpygo. We cite a powerful Premet's theorem complementary to the BorelWeilBott theorem. We mention relations with the Nijenhuis bracket and nonholonomic systems.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2004
 arXiv:
 arXiv:math/0404139
 Bibcode:
 2004math......4139L
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Differential Geometry;
 17A70;
 17B56 (Primary) 17B01;
 17B70 (Secondary)
 EPrint:
 E.Vinberg (ed.) Lie groups and invariant theory, A.L.Onishchik Festschrift, Amer.Math.Soc.Transl.Ser.2, 213, Amer.Math.Soc., Providence, RI, 2005, 157172