Note on Lie algebra kernels in characteristic 𝑝

Abstract

Let K and L be Lie algebras over a field F. Let D(K) denote the derivation algebra of K, and I(K) the ideal consisting of the inner derivations of K. If q5 is a homomorphism of L into D(K)/I(K) then 4) defines what is called the structure of an L-kernel on K. The L-kernel K is said to be extendible if there exists a Lie algebra extension with kernel K and image L which induces the given L-kernel structure on K in the natural fashion. The L-kernels with a fixed L-module C as common center can be partitioned into equivalence classes, two kernels being equivalent if they differ (in the sense of a certain composition of kernels) by an extendible kernel. It has been shown in [2] that these equivalence classes of L-kernels constitute a vector group over F which is canonically isomorphic with the 3-dimensional cohomology group H3(L, C). In particular, every L-kernel determines a 3-dimensional cohomology class which is called the obstruction of the kernel and whose vanishing is equivalent to the extendibility of the kernel. A cohomology class u for the finite dimensional Lie algebra L in the finite dimensional L-module C is said to be effaceable if there exists a finite dimensional L-module C' containing C and such that the canonical image of u in H(L, C') is 0. It is known from [2] and [3] that every effaceable 3-dimensional cohomology class for the finite dimensional Lie algebra L in a finite dimensional L-module C is the obstruction of a finite dimensional L-kernel K with center C. Moreover, in the case of characteristic 0, it has been shown in [3] that, conversely, the obstruction of a finite dimensional kernel is effaceable. The proof depends heavily on the structure and representation theory of Lie algebras of characteristic 0 and therefore breaks down completely in characteristic pH 0. The purpose of this note is to prove this result in the case of characteristic p #0. When this is combined with the known results we have just mentioned there results the following theorem for arbitrary characteristic.