A Relative Theory for Leibniz n-Algebras

TL;DR

This paper develops a new theoretical framework connecting Leibniz n-algebras and Filippov algebras, introducing co-representations and spectral sequences to advance understanding of their homological properties.

Contribution

It introduces a relative theory for Leibniz n-algebras, utilizing tensor powers and co-representations to relate different algebraic structures and derive spectral sequences.

Findings

Established anti-symmetric co-representation structures

Defined two relative theories for Leibniz n-algebras

Constructed a spectral sequence for Leibniz homology

Abstract

In this paper we show that for a $n$-Filippov algebra $\g,$ the tensor power $\g^{\otimes n-1}$ is endowed with a structure of anti-symmetric co-representation over the Leibniz algebra $\g^{\wedge n-1}$. This co-representation is used to define two relative theories for Leibniz $n$-algebras with $n>2$ and obtain exact sequences relating them. As a result, we construct a spectral sequence for the Leibniz homology of Filippov algebras.