Semiassociative algebras over a field

TL;DR

This paper introduces semiassociative algebras, a class of nonassociative algebras with an étale subalgebra acting faithfully, generalizing associative matrix algebras and forming a Brauer monoid structure.

Contribution

It defines semiassociative algebras, explores their structure as forms of skew matrices, and establishes their role within the Brauer monoid, extending classical algebraic concepts.

Findings

Semiassociative algebras generalize associative matrix algebras.

They form a Brauer monoid containing the Brauer group.

Semiassociative algebras are characterized by their nucleus containing an étale subalgebra.

Abstract

An associative central simple algebra is a form of matrices, because a maximal \'{e}tale subalgebra 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 \'{e}tale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrices, which we are led to define and investigate. Semiassociative algebras modulo skew matrices compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.