Derived brackets for fat Leibniz algebras

TL;DR

This paper develops a new derived bracket construction for fat Leibniz algebras using a 3-cocycle and Poisson algebra structures, advancing the algebraic understanding of Leibniz algebra cohomology.

Contribution

It introduces a novel derived bracket framework for fat Leibniz algebras based on cohomological and Poisson algebra techniques.

Findings

Constructed the derived bracket using a specific 3-cocycle.

Established a Poisson algebra structure on representable cochains.

Provided new insights into the cohomology of fat Leibniz algebras.

Abstract

Given a Leibniz algebra L with left center Z, we work on C(L,Z,S(Z)), the Z-standard complex of L with coefficients in S(Z). We construct the derived bracket for a fat Leibniz algebra in terms of a certain 3-cocycle and a Poisson algebra structure on the space of so-called "representable cochains".