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 $[K(R, ), K(R, *)]$ for a simple, locally finite dimensional associative algebra $R$ with involution $$. 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 $*$ such that $K(R, *) \ne [K(R, *), K(R, *)]$ to demonstrate the necessity of considering the commutator in the first two theorems.