Local inheritance in Jordan algebras of quotients

Abstract: Since their introduction by K. Meyberg in the nonassociative setting, local algebras have played a key role in the study of Jordan systems. The local inheritance of regularity conditions (such as nondegenerancy, strong primeness or primitivity) is a well-known result that undoubtedly contributed to the development of the structure theory, not only of Jordan algebras, but also of Jordan pairs and triple systems.

A rather usual strategy to tackle many Jordan questions is to differentiate Jordan systems depending on whether their local algebras satisfy or not certain properties. For instance, some of the recent results on localization theory for Jordan algebras have been established taking advantage of the dichotomy between Jordan algebras having, or not, local algebras satisfying polynomial identities. Analogously, the formulation of Goldie local theory for Jordan algebras is closely related to Jordan algebras admitting Lesieur-Croisot local algebras. The above considerations lead us to consider, in the Jordan algebra setting, how local algebras of Jordan algebras interact with their algebras of quotients (in Utumi's sense). This problem is motivated by a previous question originally posed, in the associative setting for (maximal, Martindale and symmetric) rings of quotients of semiprime rings by G\'omez Lozano and Siles Molina, who proved that both constructions commute whenever the element at which the local algebra is defined becomes von Neumann regular in the corresponding ring of quotients. In this talk we will display the Jordan algebra case of this problem, proving that, for any nondegenerate Jordan algebra, whenever the element defining the local algebra becomes von Neumann regular in its maximal algebra of quotients, taking local algebras and maximal algebras of quotients are commuting constructions. This is a joint work with Fernando Montaner (University of Zaragoza).

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar