Semi-simple Leibniz algebras I
Jörg Feldvoss
TL;DR
The paper provides a complete structural and classification framework for finite-dimensional semi-simple Leibniz algebras over fields of characteristic zero by expressing them as left hemi-semidirect products $\\mathfrak{g}\\ltimes_\\ell M$, where $\\mathfrak{g}$ is a finite-dimensional semi-simple Lie algebra and $M$ is a completely reducible anti-symmetric $\\mathfrak{g}$-bimodule with no trivial submodules. It proves precise simplicity and semi-simplicity criteria for these hemi-semidirect products and leverages Levi's theorem and Weyl's complete reducibility to reduce the classification to the well-understood theory of semi-simple Lie algebras and their irreducible modules. The results yield explicit descriptions of the Leibniz kernel, derived subalgebra, centers, and the derivation algebra as a vector space, and they establish several consequences for centers, left-centrality, and graded structure, including that right centers vanish and that semi-simple left central algebras are Lie algebras. The work also provides constructions showing that semi-simple Leibniz algebras need not decompose as direct sums of Lie-simple pieces, and it extends previous results to arbitrary characteristic-zero fields.
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.