Leibniz Representations of Lie Algebras Original Research Article

ALeibniz representationof the Lie algebra image is a vector spaceMequipped with two actions (left and right) [–, –]: imagecircle times operatorM→Mand [–, –]:Mcircle times operatorimage→Mwhich satisfy the relations[formula]when one of the variables is inMand the two others are in image. In this paper we show that the categoryL(image) of finite dimensional Leibniz representations of a finite dimensional semi-simple Lie algebra is not semi-simple, but thatL(image) has global dimension 2. We give an explicit description of the extensions of simple objects and we obtain the description of the quiver ofL(image) (in the sense of Gabriel). It turns out thatL(image) is tame only for image=sl2. We give the complete list of indecomposable objects in that case.