Derived bracket construction and anti-cyclic subcomplex of Leibniz (co)homology complex

TL;DR

This paper introduces a universal derived bracket representation for Leibniz algebras, showing it forms a subcomplex of Leibniz (co)homology and relates to the anti-cyclicity of the Leibniz operad.

Contribution

It constructs the universal derived bracket representation and demonstrates its role as a subcomplex reflecting the anti-cyclicity property of Leibniz operads.

Findings

Universal derived bracket representation constructed

Subcomplex of Leibniz (co)homology identified

Connection to anti-cyclicity of Leibniz operad established

Abstract

An arbitrary Leibniz algebra can be embedded in a differential graded Lie algebra via the derived bracket construction. Such an embedding is called a derived bracket representation. We will construct the universal version of the derived bracket representation, prove that the principal part of the target dg Lie algebra defines a subcomplex of Leibniz (co)homology complex and that the existence of the subcomplex is a reflection of the anti-cyclicity of the Leibniz operad.