Leibniz algebras of Heisenberg type

TL;DR

This paper introduces and classifies a new class of Leibniz algebras related to Heisenberg algebras, focusing on their module structures and providing a detailed classification for specific cases.

Contribution

It provides a classification theorem for Heisenberg-Fock Leibniz algebras and explores their module structures, including generalizations and minimal faithful representations.

Findings

Classification of Heisenberg-Fock Leibniz algebras

Analysis of Leibniz algebras with H_3 as Lie algebra

Description of Leibniz algebras with minimal faithful H_n action

Abstract

We introduce and provide a classification theorem for the class of Heisenberg-Fock Leibniz algebras. This category of algebras is formed by those Leibniz algebras $L$ whose corresponding Lie algebras are Heisenberg algebras $H_n$ and whose ${H_n}$-modules $I$, where $I$ denotes the ideal generated by the squares of elements of $L$, are isomorphic to Fock modules. We also consider the three-dimensional Heisenberg algebra $H_3$ and study three classes of Leibniz algebras with $H_3$ as corresponding Lie algebra, by taking certain generalizations of the Fock module. Moreover, we describe the class of Leibniz algebras with $H_n$ as corresponding Lie algebra and such that the action $I \times H_n \to I$ gives rise to a minimal faithful representation of $H_n$. The classification of this family of Leibniz algebras for the case of $n=3$ is given.