On $R$-equivalence of Automorphism Groups of Associative Algebras

TL;DR

This paper investigates the rationality and $R$-equivalence properties of the automorphism groups of finite-dimensional associative algebras over arbitrary fields, focusing on the identity component over perfect fields.

Contribution

It provides new insights into the $R$-equivalence of automorphism groups of associative algebras, extending understanding over arbitrary fields.

Findings

Analysis of $R$-equivalence for automorphism groups over perfect fields

Characterization of the rationality properties of automorphism groups

Extension of known results to broader classes of fields

Abstract

Let $A$ be a finite-dimensional associative $k$-algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all $k$-algebra automorphisms of $A$, as an affine algebraic group over an arbitrary field $k$. We investigate mainly the $R$-equivalence property of the identity component of $\mathrm{Aut}_{k}(A)$ over a perfect field $k$.