Pairs of quotients of Jordan pairs

Abstract: In this talk we expose ongoing joint work with I Paniello on systems of quotients (in a sense partially extending the localization theory of Jordan algebras, which in turn is inspired by the localization theory of associative algebras). Localization theory in associative algebras originated in the purpose of extending the construction of fields of quotients of integral domains, and therefore in the purpose of defining ring extensions in which a selected set of elements become invertible. As it is well known in associative theory that led to Goldie's theorems, and these in turn to more general localization theories for which the denominators of the fraction-like elements of the extensions are (one-sided) ideals taken in a class of filters (Gabriel filters). These ideas have been partially extended to Jordan algebras by several authors (starting with Zelmanov's version of Goldie theory in the Jordan setting, and its extension by Fernandez López-García Rus and Montaner) and Paniello and Montaner (among others) definition of algebras of quotients of Jordan algebras. Following the development of Jordan theory, a natural direction for extending these results is considering the context of Jordan pairs. This is the objective of the research presented here. Since obviously a Jordan pair cannot have invertible elements unless it is an algebra, and in this case we are back in the already developed theory, the kind of quotients that would make a significative (proper) extension of the case of algebras should be based in a different notion of quotient. An approach that seems to be promising is considering the Jordan extension of Fountain and Gould notion of local order, as has been adapted to Jordan algebras by the work of Fernández López, and more recently by Montaner and Paniello with the notion of local order, in which the bridge between algebras and pairs is established by local algebras following the ideas of D'Amour and McCrimmon. In the talk this idea is exposed, together with the state of the research, and the open problems that it raises.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar