Isotopisms of nilpotent Leibniz algebras and Lie racks

Abstract

Leibniz algebras were introduced by J.-L. Loday in 1993 as a non-antisymmetric version of Lie algebras. Many results of Lie algebras have been extended to Leibniz algebras. One of them is the Levi decomposition, which states that every finite-dimensional Leibniz algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra. This makes clear the importance of the problem of Lie / Leibniz algebra classification, which has been dealt with since the early 20th century. However in general, given two Leibniz algebras g and h, it is hard to check if g and h are isomorphic or not, but it is easier to see if there exists an isotopism between them, i.e. if there is a triple of linear isomorphisms (f, g, h): g --> h such that [ƒ(x),g(y)]=h([x,y]), for any x,y in g. The notion of isotopism between two algebraic structures was explicitly introduced in 1942 by Abraham Adrian Albert in order to classify non- associative algebras and we can use it in the case of Leibniz algebras. In this talk we study the isotopism classes of two-step nilpotent Leibniz algebras. We show that every nilpotent Leibniz algebra with one-dimensional commutator ideal is isotopic to the Heisenberg algebra or to the Heisenberg Leibniz algebra l_J1 where J1 is the n ×n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification, up to isotopism, of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with commutator ideal of dimension one. Eventually we introduce new isotopism invariants for Leibniz algebras and Lie racks. Joint work with Gianmarco La Rosa and Gábor P. Nagy.