PBW for an inclusion of Lie algebras

TL;DR

This paper establishes a criterion for when a PBW-type isomorphism exists for inclusions of Lie algebras, generalizing classical results and linking Lie algebra theory with algebraic geometry.

Contribution

It provides a necessary and sufficient condition for splitting the filtration in Lie algebra inclusions, extending the classical PBW theorem to more general settings.

Findings

A criterion for the existence of a splitting of the filtration.

Recovery of the classical PBW theorem for diagonal embeddings.

Connection between Lie algebra filtrations and algebraic geometry results.

Abstract

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(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 h-modules U(g)/U(g)h and S(n). For the diagonal embedding h \subset h \oplus h the condition is automatically satisfied and we recover the classical Poincae-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.