A non-abelian exterior product and homology of Leibniz algebras

TL;DR

This paper introduces a non-abelian exterior product for Leibniz algebras, explores its connection to homology, and applies it to construct exact sequences and compare homologies of Lie and Leibniz algebras.

Contribution

It presents a novel non-abelian exterior product for Leibniz algebras and links it to homology and quadratic functors, advancing the understanding of Leibniz homology.

Findings

Established the relation between the exterior product and Leibniz homology.

Constructed an eight-term exact sequence in Leibniz homology.

Compared second Lie and Leibniz homologies using the new exterior product.

Abstract

We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.