Lie Algebras with Nilpotent Centralizers

Abstract

We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.