Some Leibniz bimodules of 𝔰𝔩2

Abstract

We study complex finite-dimensional Leibniz algebra bimodule over [Formula: see text] that as a Lie algebra module is split into a direct sum of two simple [Formula: see text]-modules. We prove that in this case there are only two nonsplit Leibniz [Formula: see text]-bimodules and we describe the actions.

Finite-dimensional Leibniz algebra representations of $\mathfrak{sl}_2$

“…This work is a direct continuation of an investigation started in [7]. If M is an irreducible Leibniz representation of sl 2 , by Weyl's result the left action on M as a Lie algebra representation decomposes into a direct sum of irreducible Lie representations of sl 2 .…”

“…Hence, the problem of description reduces to the study of the right action. In case the number of such irreducible Lie representations is two, up to a Leibniz algebra representation isomorphism there are exactly two types of irreducible Leibniz representations, whose actions are described in [7,Theorem 3.1]. In the current work, we establish the description in full generality providing an explicit description of actions up to isomorphism in Theorem 3.5.…”

“…A bimodule is called indecomposable, if it is not a direct sum of its subbimodules. Obviously, a simple bimodule is indecomposable, while the converse is not necessarily true (see [7,Proposition 1.2] and a paragraph that follows). To study bimodules it suffices to study indecomposable ones.…”

“…In the current work, we use results on the left action established in [7] using only equality (2.2). Till the rest of the section let M be an sl 2 -bimodule that decomposes as a right module into the direct sum…”

“…[3,8]), the representation of a Leibniz algebra is given in [9] as a K-module M with two actions -left and right, satisfying compatibility conditions coming from a so called square-zero construction. It is known that the category of Leibniz representations of a given Leibniz algebra is not semisimple and any non-Lie Leibniz algebra admits a representation, which is neither simple, nor completely reducible [7,Proposition 1.2]. In [10] the indecomposable objects of the category of Leibniz representations of a Lie algebra are studied and for sl 2 the indecomposable objects in that category are described as extensions (see Theorem 2.5 below).…”