Semi-simple Leibniz algebras I

Abstract

The goal of this paper is to describe the structure of finite-dimensional semi-simple Leibniz algebras in characteristic zero. Our main tool in this endeavor are hemi-semidirect products. One of the major results of this paper is a simplicity criterion for hemi-semidirect products. In addition, we characterize when a hemi-semidirect product is semi-simple or Lie-simple. Using these results we reduce the classification of finite-dimensional semi-simple Leibniz algebras over fields of characteristic zero to the well-known classification of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. As one consequence of our structure theorem, we determine the derivation algebra of a finite-dimensional semi-simple Leibniz algebra in characteristic zero as a vector space. This generalizes a recent result of Ayupov et al. from the complex numbers to arbitrary fields of characteristic zero.