Le laplacien d'une quasi-bialgèbre de Lie

Abstract

Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any finite-dimensional Lie quasi-bialgebra structure $(\mathcal{G}, \mu, \gamma, \phi)$ and a $\mathcal{D}$-module structure $M$, where $\mathcal{D}$ is the double of the given Lie quasi-bialgebra, we associate one operator $L_{M} =\partial_{\mu, M}d_{\gamma, M} + d_{\gamma, M}\partial_{\mu, M}$ called the laplacien of the Lie quasi-bialgebra associated to the $\mathcal{D}$-module structure. We establish the fondamentals properties of the laplacian and give an explicit formula for $L_{M}$ by mean of adjoint characters of $\mathcal{G}$ and $\mathcal{G^*}$.