Algebras of quotients and Martindale-like quotients of Leibniz algebras

TL;DR

This paper introduces and studies algebras of quotients and Martindale-like quotients for Leibniz algebras, exploring their properties, maximal quotients, and relationships with associative algebras, advancing the structural understanding of Leibniz algebras.

Contribution

It defines new quotient concepts for Leibniz algebras, constructs maximal quotients for semiprime cases, and examines their connections with associative algebras and dense extensions.

Findings

Properties of Leibniz algebras extend to their quotients

Maximal algebra of quotients constructed for semiprime Leibniz algebras

Relationship established between Leibniz quotients and associative algebras

Abstract

In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by left and right multiplication operators of the corresponding Leibniz algebras of quotients are examined. Finally, the definition of dense extensions and some vital properties about Leibnia algebras via dense extensions are introduced.