Simple, locally finite dimensional Lie algebras in positive characteristic

Abstract

We prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic $p$ which give sufficient conditions for the algebras to be of the form $[R^{(-)},R^{(-)}]/(Z(R)\cap [R^{(-)},R^{(-)}])$ or $[K(R,\ast ),K(R,\ast )]$ for a simple, locally finite dimensional associative algebra $R$ with involution $\ast$. The first proves that a condition we introduce, known as locally nondegenerate, along with the existence of an ad-nilpotent element suffice. The second proves that a uniformly ad-integrable Lie algebra is of this type if the characteristic of the ground field is sufficiently large. Lastly we construct a simple, locally finite dimensional associative algebra $R$ with involution $\ast$ such that $K(R,\ast )\ne [K(R,\ast ),K(R,\ast )]$ to demonstrate the necessity of considering the commutator in the first two theorems.

arXiv comment17 pages