On $R$-triviality of $F_4$

TL;DR

This paper proves that the automorphism groups of Albert division algebras arising from the first Tits construction are $R$-trivial, contributing to the understanding of algebraic group properties over fields.

Contribution

It establishes the $R$-triviality of automorphism groups of Albert division algebras from the first Tits construction, a case previously unexplored.

Findings

Automorphism groups of Albert division algebras are $R$-trivial.

The result applies specifically to Albert algebras from the first Tits construction.

Advances understanding of algebraic groups over fields in relation to $R$-triviality.

Abstract

It is known that simple algebraic groups of type $F_4$ defined over a field $k$ are precisely the full automorphism groups of Albert algebras over $k$. We explore $R$-triviality for the group $\text{\bf Aut}(A)$ when $A$ is an Albert algebra. In this paper, we consider the case when $A$ is an Albert division algebra, that arises from the first Tits construction. We prove that $\text{\bf Aut}(A)$ is $R$-trivial, in the sense of Manin.