Depth Generated Simple Lie Algebras

Abstract

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra Open image in new window of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra Open image in new window. This determines a depth functor Open image in new window from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras Open image in new window corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras Open image in new window are simple for all i ∈ ω. In particular, the Open image in new window (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras.