Associative and Lie algebras of quotients

Abstract

In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra Q is an algebra of quotients of a Lie algebra L in terms of the associative algebras generated by the adjoint operators of L and Q respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.

Strongly non-degenerate Lie algebras

“…Examples of extensions where Ann L (Q) = 0 are the dense ones (see [3] for the definition of a dense extension and [14] for examples).…”

“…In the seminal paper [15] the second author initiated the study of algebras of quotients of Lie algebras, by adapting some ideas from the associative and also Jordan ( [13]) contexts. She introduced the notion of a general (abstract) algebra of quotients of a Lie algebra, and also the notion of the maximal algebra of quotients Q m (L) of a semiprime Lie algebra L. Follow-up results can be found in [14,4,2].…”

Algebras of quotients of Hom-Lie algebras

“…In [15], authors examined how the notion of algebras of quotients for Lie algebras tied up with the corresponding well-known concept in the associative case. Inspired by the method in [15], we mainly study the relationship between Hom-Lie algebras and the associative algebras generated by inner derivations of the corresponding Hom-Lie algebras of quotients. First of all, we will give some definitions and basic notations.…”

“…As we know, one popular topic of research is the relationship between algebras of quotients of non-associative algebras and associative algebras in recent years. In [15], authors examine the relationship between associative and Lie algebras of quotients. Inspired by them, we'll explore whether there exists a relationship between associative and Hom-Lie algebras of quotients.…”

Algebras of quotients and Martindale-like quotients of Leibniz algebras

“…As we know, one popular topic of research is the relationship between algebras of quotients of non-associative algebras and associative algebras in recent years. In [21], authors examined the relationship between associative and Lie algebras of quotients. Inspired by them, we'll explore whether there exists a relationship between associative and Leibniz algebras of quotients.…”