Azumaya Algebras Without Involution

TL;DR

This paper explores the properties of Azumaya algebras without involution, demonstrating limitations on degrees of involution-admitting algebras and contrasting involution types in split and non-split cases.

Contribution

It constructs specific Azumaya algebras showing minimal degrees for involution existence and provides examples distinguishing symplectic and orthogonal involutions.

Findings

Constructed degree 4 Azumaya algebra with involution degree divisible by 8.

Examples of degree 2 Azumaya algebras with symplectic but no orthogonal involutions.

Contrasts with central simple algebras over fields regarding involution types.

Abstract

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a $2$-torsion class in the Brauer group if and only if there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that $2\mathrm{deg}\, A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$ such that the degree of any algebra $A'$ in the Brauer class of $A$ admitting an involution is divisible by $8$. Separately, we provide examples of split and non-split Azumaya algebras of degree $2$ admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.