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

Source: [Derived bracket construction and anti-cyclic subcomplex of Leibniz (co)homology complex](<https://www.sota2.com/research/paper/68fa50a21e209131ce96d37e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T10:58:30.607Z

Authors: K. Uchino

Published: 2013-12-27T14:27:50.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1312.7268>)

[PDF](<https://arxiv.org/pdf/1312.7268.pdf>)