Représentation du double d’une quasi-bigèbre de Lie dans son algèbre extérieure

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional \(({\mathcal {G}}, \mu , \gamma , \phi )\), correspond a unique Lie algebra structure on \({\mathcal {D}}= {\mathcal {G}}\oplus \mathcal {G^{*}}\) which leaves invariant the canonical scalar product on \({\mathcal {D}}\), called the double of the given Lie quasi-bialgebra. We show that there exist on \(\Lambda {\mathcal {G}}\), the exterior algebra of \({\mathcal {G}}\), a \({\mathcal {D}}\)-module structure.