Two-step nilpotent Leibniz algebras

Leibniz algebras were rst 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 signi cant role in di erent areas of mathematics and physics. In this talk we give the classi cation of two-step nilpotent Leibniz algebras over a eld 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 nxn matrix A in canonical form, the Kronecker Leibniz algebras and the Dieudonne Leibniz algebras, both parametrized by their dimension only. Then we describe the Lie algebras of derivations of this class of Leibniz algebras and we show that every almost inner derivation of a nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation. 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.