PBW for an inclusion of Lie algebras
Résumé
Let $\mathfrak{h}\subset \mathfrak{g}$ be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the $\mathfrak{h}$-module $U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{h}$ whose associated graded $\mathfrak{h}$-module is isomorphic to $S(\mathfrak{n})$. We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the $\mathfrak{h}$-modules $U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{h}$ and $S(\mathfrak{n})$. For the diagonal embedding $\mathfrak{h} \subset \mathfrak{h} \oplus \mathfrak{h}$ the condition is automatically satisfied and we recover the classical Poincaré-Birkhoff-Witt theorem. The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way.