Leibniz algebras were introduced by Blokh in the 1960s and rediscovered by Loday in the 1990s as non-anticommutative analogues of Lie algebras. Many results for Lie algebras have been proved to hold for Leibniz algebras, but there are also several results that are not true in this more general context. In my talk, I will investigate the structure of semi-simple Leibniz algebras. In particular, I will prove a simplicity criterion for (left) hemi-semidirect products of a Lie algebra 𝔤 and a (left) 𝔤-module. For example, in characteristic zero every finite-dimensional simple Leibniz algebra is such a hemi-semidirect product. But this also holds for some infinite-dimensional Leibniz algebras or sometimes in non-zero characteristics. More generally, the structure of finite-dimensional semi-simple Leibniz algebras in characteristic zero can be reduced to the well-known structure of finite-dimensional semi-simple Lie algebras and their finite-dimensional irreducible modules. If time permits, I will apply these structure results to derive some properties of finite-dimensional semi-simple Leibniz algebras in characteristic zero and other Leibniz algebras that are hemi-semidirect products.
This video is part of the European Non-Associative Algebra Seminar series.
