Depth Generated Simple Lie Algebras

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 L-D of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra A(D). This determines a depth functor D : (A, L) -> (A(D), L-D) from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras A(D) L-D 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 A(Di) L-Di are simple for all i is an element of omega. In particular, the A(Di) L-Di (i is an element of omega.) corresponding to central simple Jordan Lie algops (A, L) are simple Lie algebras.