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 Back 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 x 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. (C) 2016 Elsevier Inc. All rights reserved.