Injective hulls of Lie modules (Q801016)

Let L be a finite dimensional Lie algebra over an algebraically closed field k. Denote by \(E_ L(k)\), the injective hull of the 1-dimensional trivial L-module. The goal of this paper is to describe \(E_ L(k)\) where L is the semi-direct product \(N\rtimes H\) of a nilpotent Lie algebra N by a Lie algebra H. Theorem. \(E_ L(k)\cong E_ N(k)\otimes_ k E_ H(k)\) as L-modules. A nice observation that follows as a corollary is that \(E_ L(k)\) is locally finite dimensional for every solvable Lie algebra L. The paper gives complete proofs and ends with several illustrative examples.