Simple Lie Algebras of Special Type,

Let K be a field, let A be an associative, commutative K-algebra, and let Δ be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, (A, Δ) = A KΔ = AΔ becomes a Lie algebra, a Witt type algebra. In addition, there is a map div: (A, Δ) → A called the divergence and its kernel S = (A, Δ) is a Lie subalgebra, a special type algebra. In this paper, we study S from a ring theoretic point of view, obtaining sufficient conditions for the Lie simplicity of [S, S]. While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas are of interest in their own right and may, in time, lead to a more satisfactory answer.