Algèbres de Poisson et algèbres de Lie résolubles

Abstract

Let $\mathfrak{g}$ be a solvable Lie algebra and $Q$ an $ad\mathfrak{g}$-stable prime ideal of the symmetric algebra $S(\mathfrak{g})$ of $\mathfrak{g}$. If $E$ is the set of non zero elements of $S(\mathfrak{g})/Q$ which are eigenvectors for the adjoint action of $\mathfrak{g}$ in $S(\mathfrak{g})/Q$, the localised algebra $(S(\mathfrak{g})/Q{)}_E$ has a natural structure of Poisson algebra. We study this algebra here.

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