Automorphisms of Albert algebras and a conjecture of Tits and Weiss

TL;DR

This paper proves the Tits-Weiss conjecture for Albert division algebras arising from pure first Tits constructions over any field, linking it to the Kneser-Tits problem for certain forms of E8 and establishing R-triviality of their automorphism groups.

Contribution

It establishes the Tits-Weiss conjecture for a class of Albert division algebras and connects it to the Kneser-Tits problem for specific E8 forms, providing new insights into algebraic group structures.

Findings

Proof of the Tits-Weiss conjecture for pure first Tits construction Albert division algebras.

Solution to the Kneser-Tits problem for certain forms of E8 related to Albert algebras.

Demonstration that the automorphism group of such Albert algebras is R-trivial.

Abstract

Let $k$ be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over $k$ which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra $A$ over a field $k$, every norm similarity of $A$ is inner modulo scalar multiplications. It is known that $k$-forms of $E_8$ with index $E^{78}_{8,2}$ and anisotropic kernel a strict inner $k$-form of $E_6$ correspond bijectively (via Moufang hexagons) to Albert division algebras over $k$. The Kneser-Tits problem for a form of $E_8$ as above is equivalent to the Tits-Weiss conjecture (see \cite{TW}). Hence we provide a solution to the Kneser-Tits problem for forms of $E_8$ arising from pure first Tits construction Albert division algebras. As an application, we prove that for $G={\bf Aut}(A),~G(k)/R=1$, where $A$ is a pure first construction Albert division algebra over $k$ and $R$ stands for $R$-equivalence in the sense of Manin (\cite{M}).