Omni-representations of Leibniz algebras

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