Azumaya Algebras Without Involution
Asher Auel, Uriya A. First, Ben Williams

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 over a ring represents a -torsion class in the Brauer group if and only if there is an algebra in the Brauer class of admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose such that . We show that is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra of degree and period such that the degree of any algebra in the Brauer class of admitting an involution is divisible by . Separately, we provide examples of split and non-split Azumaya algebras of degree 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…
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