Two-step nilpotent Leibniz algebras

Leibniz algebras were first introduced by J.-L. Loday as a non-antisymmetric version of Lie algebras, and many results of Lie algebras have been extended to Leibniz algebras. Earlier, such algebraic structures had been considered by A. Blokh, who called them D-algebras. Leibniz algebras play a significant role in different areas of mathematics and physics. Lie algebras with small dimensional commutator ideals have been considered. The purpose of this talk is to classify the two-step nilpotent Leibniz algebras over a field of characteristic different from 2 in terms of Kronecker modules associated with pairs of bilinear forms. We show that there are only three classes of nilpotent Leibniz algebras with one-dimensional commutator ideal. Up to isomorphisms, the first class is determined by the companion matrix of the power of a monic irreducible polynomial, and the other two are unique. In this way, we give the definition of Heisenberg Leibniz algebra, Kronecker Leibniz algebra and Dieudonné Leibniz algebra. Then we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n + 1)−dimensional Heisenberg Lie algebra. Moreover, when the dimension is 3, we determine all the isomorphism classes of these algebras. This is joint work with Gianmarco La Rosa (University of Palermo).