Two-step nilpotent Leibniz algebras

TL;DR

This paper classifies two-step nilpotent Leibniz algebras using Kronecker modules, explores their real and complex forms, and demonstrates their integration into Lie structures, extending classical Lie algebra concepts.

Contribution

It provides a complete classification of two-step nilpotent Leibniz algebras and establishes their global integration into Lie group structures, generalizing classical results.

Findings

Complete classification of two-step nilpotent Leibniz algebras

Existence of global integration for nilpotent real Leibniz algebras

Every Lie quandle integrating a Leibniz algebra is induced by conjugation in a Lie group

Abstract

In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, 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 $\mathfrak{h}_{2n+1}$. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. We also 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.