Representations of Leibniz algebras (Q2348556)

This paper is devoted to the study of irreducible representations of Leibniz algebras. The authors establish a result which claims that irreducible Leibniz representations are very closely related to irreducible representations of the corresponding Lie algebra. Namely, for a Leibniz bimodule in order to be simple it is necessary either to admit a trivial left action (in which case the right action differs from the Lie action up to a sign), or the left action must differ by a sign from the right one. Moreover, in the case of semi-simple Leibniz algebras due to the existence of decomposition of such algebra into a semi-direct sum of its solvable radical and an ideal called Leibniz kernel, the authors obtain a further simplification on the nature of Leibniz representations. Left and right actions turn out to act trivially on the Leibniz kernel.