Omni-Representations of Leibniz Algebras

Abstract

In this paper, first we introduce the notion of an omni-representation of a Leibniz algebra $g$ on a vector space $V$ as a Leibniz algebra homomorphism from $g$ to the omni-Lie algebra $gl(V)⊕V.$ Then we introduce the omni-cohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.