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. Nowdays Leibniz algebras play a significant role in different areas of mathematics and physics. In this talk we give the classification of two-step nilpotent Leibniz algebras over a field F 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, which we call the Heisenberg Leibniz algebras, parametrized by the dimension 2n+ 1 and a n×n matrix A in canonical form, the Kronecker Leibniz algebras and the Dieudonné Leibniz algebras, both parametrized by their dimension only. Moreover, using the Leibniz algebras / Lie local racks correspondence, we show that nilpotent real Leibniz algebras have always a global integration. As an example, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group. This is joint work with Gianmarco La Rosa (University of Palermo).