Séminaire d'algèbre et de géométrie du 01-07-2024

Exposé de Uzi Vishne

Le 01-07-2024 à 15:00, en P108 et en ligne.

Semiassociative Algebras over a Field

Résumé

An associative central simple algebra is a form of a matrix algebra, because a maximal etale subalgebra, which splits over the algebraic closure, acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an etale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. (Some families of nonassociative of cyclic algebras, studied by Pumplun and others, naturally fall into this new class.) Semiassociative algebras modulo skew matrix algebras compose a “nonassociative Brauer monoid”, which contains the classical Brauer group of the field as a unique maximal subgroup. This report is on joint work with Blachar, Haile, Matzri, and Rein.