Extensions of representations of Lie algebras

John Gerard Ryan

Vol. 127 (1987), No. 1, 173–186

DOI: 10.2140/pjm.1987.127.173

Abstract

Let ϕ : L₁ → L₂ be a morphism of finite-dimensional Lie algebras over a field of characteristic zero. Our problem is this: given a finite-dimensional L₁-module, V say, when does V embed as a sub L₁-module of some finite-dimensional L₂-module? The problem clearly reduces to the case in which ϕ is injective. We provide here (Thm. 3.6) a solution in two separate cases: (i) under the assumption that ϕ maps the radical of L₁ into the radical of L₂, or (ii) under the assumption that L₁ is its own commutator ideal.

Mathematical Subject Classification 2000

Primary: 17B10

Milestones

Received: 15 November 1984

Published: 1 March 1987