Leibniz representations of Lie algebras

A Leibniz representation of the Lie algebra g is a vector space M equipped with two actions (left and right) [-, -]: g x M --> M and [-, -]: M x g --> M which satisfy the relations [x,[y,z]] = [[x,y],z] - [[x,z],y], when one of the variables is in M and the two others are in g. In this paper we show that the category L(g) of finite dimensional Leibniz representations of a finite dimensional semi-simple Lie algebra is not semi-simple, but that L(g) has global dimension 2. We give an explicit description of the extensions of simple objects and we obtain the description of the quiver of L(g) (in the sense of Gabriel). It turns out that L(g) is tame only for g = Sl(2). We give the complete list of indecomposable objects in that case. (C) 1996 Academic Press. Inc.