The structure of injective hulls of Lie modules.

Abstract

Let N be a finite dimensional nilpotent Lie algebra over an algebraically closed field k of characteristic zero. In this paper we show that the injective hull of the 1-dimensional trivial U(N)-module (here U(N) denotes the universal enveloping algebra of N) is isomorphic to the k-algebra of polynomials in n indeterminates, n = dim(, k)(N), with N acting as a Lie algebra of derivations. These derivations can be expressed as sums of partial derivatives with polynomial coefficients whose degrees are bounded above by d-1 where d is the index of nilpotency of N.

Assume now that L is a finite dimensional Lie algebra over k which can be expressed as the semi-direct product of a nilpotent ideal N and a subalgebra H. Then the injective hull of the 1-dimensional trivial U(L)-module is isomorphic to the tensor product over k of the injective hulls of the 1-dimensional trivial U(N)-module and the 1-dimensional trivial U(H)-module, the latter two modules being equipped with suitable L-module structures. This result is used to show that the injective hull of a locally finite dimensional module over a finite dimensional solvable Lie algebra over k is locally finite dimensional. Furthermore, if L is the semi-direct product of a finite dimensional nilpotent ideal and a finite dimensional abelian subalgebra, then the injective hull of the 1-dimensional trivial U(L)-module is isomorphic to the k-algebra of polynomials in m indeterminates where m = dim(, k)(L). L acts as a Lie algebra of derivations and these derivations can be expressed as sums of partial derivatives with polynomial coefficients. A method of calculating these representations is indicated and a number of examples are exhibited.