Leibniz algebras of Heisenberg type

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 Hn and whose Hn-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 H3 and study three classes of Leibniz algebras with H3 as corresponding Lie algebra, by taking certain generalizations of the Fock module. Moreover, we describe the class of Leibniz algebras with Hn as corresponding Lie algebra and such that the action I × Hn ! I gives rise to a minimal faithful representation of Hn. The classification of this family of Leibniz algebras for the case of n = 3 is given.