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arXiv · 1411.3861

Leibniz algebras of Heisenberg type

A. J. Calderón·L. M. Camacho·B. A. Omirov
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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.

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A. J. Calderón, L. M. Camacho, B. A. Omirov. 2014-11-14. Leibniz algebras of Heisenberg type. https://arxiv.org/abs/1411.3861

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