Quasi-big\`ebres de Lie et cohomologie d'alg\`ebre de Lie

TL;DR

This paper explores the structure of Lie quasi-bialgebras, generalizing Lie bialgebras, and establishes an isomorphism between certain module structures related to their doubles and exterior algebras.

Contribution

It introduces a D-module structure on the exterior algebra of G and proves an isomorphism with the endomorphisms of the exterior algebra, deepening understanding of Lie quasi-bialgebra representations.

Findings

Existence of a D-module structure on mbda G

Isomorphism between mbda D and End(mbda G) as D-modules

Extension of Lie bialgebra theory to quasi-bialgebras

Abstract

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional (G, \mu, \gamma ,\phi ?), correspond one Lie algebra structure on D = G\oplus G*, called the double of the given Lie quasi-bialgebra. We show that there exist on \Lambda G, the exterior algebra of G, a D-module structure and we establish an isomorphism of D-modules between \Lambda D and End(\Lambda G), D acting on \Lambda D by the adjoint action.