$\delta$-Leibniz algebras and related $\delta$-type algebras

TL;DR

This paper explores $oldsymbol{ ext{delta}}$-Leibniz algebras, a parametric generalization of Leibniz algebras, defining related algebraic structures and analyzing their properties and connections to nilalgebras within a unified framework.

Contribution

It introduces $oldsymbol{ ext{delta}}$-Leibniz and related algebras, establishing their properties and relationships, thus broadening the understanding of non-associative algebra classes through a parametric approach.

Findings

Defined $oldsymbol{ ext{delta}}$-Lie, $oldsymbol{ ext{delta}}$-Lie dialgebras, and $oldsymbol{ ext{delta}}$-Zinbiel algebras.

Established connections between $oldsymbol{ ext{delta}}$-Leibniz algebras and nilalgebras.

Provided a unified framework for various non-associative algebras using the $oldsymbol{ ext{delta}}$ parameter.

Abstract

This paper introduces and investigates the structure of $\delta$-Leibniz algebras, which serve as a parametric generalization of classical Leibniz algebras defined by a scalar $\delta$. The authors define $\delta$-Lie algebras, $\delta$-Lie dialgebras, and $\delta$-Zinbiel algebras via a standard procedure and study their fundamental properties. Furthermore, the research describes symmetric $\delta$-Leibniz algebras and algebras of $\delta$-biderivation type, establishing their connections with nilalgebras. Finally, these results provide a unified framework for understanding various classes of non-associative algebras through the lens of the $\delta$ parameter.