PBW for an inclusion of Lie algebras
arXiv:1010.0985 · doi:10.1016/j.jalgebra.2012.12.008
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.
Major revision, proofs of several results rewritten. Added a section explaining the case of a general representation, as opposed to the trivial one. 20 pages, LaTeX
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