Enveloping lie algebras of Leibniz algebras and associated lie group-like objects

Leibniz algebras are generalization of well-known Lie algebras. Lie algebra has its Lie group, most of the applications of Lie algebras go through the Lie groups. For Leibniz algebras Lie group-like object was not found and it is of great interest. In this paper we turn in two constructions of Lie algebras (the enveloping Lie algebras of Leibniz algebras) closely related to Leibniz algebras. One hopes the Lie group-like objects for Leibniz algebras can be constructed as some combinations of Lie groups of the Lie algebras raised. We describe the enveloping Lie algebras of Leibniz algebras of dimensions two and three. These results can be used to associate geometric objects (loop spaces) to low dimensional Leibniz algebras.