Central Simple Algebras with Involution: A Geometric Approach

TL;DR

This paper extends the geometric correspondence between central simple algebras and $PGL_n$-varieties to include algebras with involution, analyzing the associated varieties with additional group actions and involution properties.

Contribution

It generalizes the category equivalence to central simple algebras with involution using geometric methods and describes involution properties via group actions on associated varieties.

Findings

Established the action of $P_{n, au}$ on associated varieties

Characterized involution properties through stabilizers in general position

Extended the geometric correspondence to involution cases

Abstract

Let $k$ be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free $PGL_n$-varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of the semidirect product $P_{n,\tau}:=PGL_n\rtimes<\tau>$, where $\tau$ is the automorphism of $\PGLn$ given by $\tau(h)=(h^{-1})^transpose$. Basic properties of an involution are described in terms of the action of $P_{n,\tau}$ on the associated variety, and in particular in terms of the stabilizer in general position for this action.