Systems of Quotients of Lie Triple Systems

Abstract

In this paper, we introduce the notion of system of quotients of Lie triple systems and investigate some properties which can be lifted from a Lie triple system to its systems of quotients. We relate the notion of Lie triple system of Martindale-like quotients with respect to a filter of ideals and the notion of system of quotients, and prove that the system of quotients of a Lie triple system is equivalent to the algebra of quotients of a Lie algebra in some sense, and these allow us to construct the maximal system of quotients for nondegenerate Lie triple systems.

Generalized derivations of Lie triple systems

“…Definition 2.1 ( [16]). A Lie triple system is a pair .T; OE ; ; / consisting of a vector space T over a field F, a trilinear multiplication OE ; ; W T T T !…”

On split Lie color triple systems

“…One may suppose that [Tα , T−α] ≠ and {Tα , T−α , T β } ≠ . By the de nition of Lie color triple systems, one has {T−α , T β , Tα} ≠ or {T β , Tα , T−α} ≠ , contradicting Lemma 3.1(4). Hence, {Tα , T−α , T β } = .…”

“…The role played by Lie triple systems in the theory of symmetric spaces is parallel to that of Lie algebras in the theory of Lie groups: the tangent space at every point of a symmetric space has the structure of a Lie triple system. Because of close relation to Lie algebras and theoretical physics, Lie triple systems have recently been widely studied [1][2][3][4]. The notion of Lie color algebras was introduced as generalized Lie algebras in 1960 by Ree [5].…”

Algebras of Quotients of Jordan–Lie Algebras

“…Utumi researched a maximal left quotient ring Q l max R and constructed it. Maximal in the sense that every left quotient ring of R can be included in Q l max R via a monomorphism which is the identity when restricted to R. Our work is based on [1] and [3] and our aim is to introduce the notion of algebra of quotients of a Jordan-Lie algebra, study its properties and construct a maximal algebra of quotients of a semiprime Jordan-Lie algebra. First, we introduce some basic definitions and properties of Jordan-Lie algebras.…”