Leibniz algebras of Heisenberg type

We introduce and provide a classification theorem for the class of Heisenberg-Fock Leibniz algebras. This type of algebras is formed by those Leibniz algebras L whose corresponding Lie algebras are isomorphic to 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 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 H_n as corresponding Lie algebra and such that the action I×H_n→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.