Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1nn,1. We introduce a Fock module for the algebra nn,1nn,1 and provide classification of Leibniz algebras LL whose corresponding Lie algebra L/IL/I is the algebra nn,1nn,1 with condition that the ideal II is a Fock nn,1nn,1-module, where II is the ideal generated by squares of elements from LL.We also consider Leibniz algebras with corresponding Lie algebra nn,1nn,1 and such that the action I×nn,1→II×nn,1→I gives rise to a minimal faithful representation of nn,1nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n=4n=4.