Albert algebras and the Tits-Weiss conjecture

TL;DR

This paper proves the Tits-Weiss conjecture for Albert division algebras over any field characteristic, establishing that all norm similarities are generated by scalar homotheties and $U$-operators, and confirms the Kneser-Tits conjecture for certain algebraic groups.

Contribution

It demonstrates the Tits-Weiss conjecture for Albert division algebras and proves the Kneser-Tits conjecture for specific algebraic groups over arbitrary fields.

Findings

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

Establishment of $R$-triviality for certain algebraic groups.

Validation of the Kneser-Tits conjecture for groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$.

Abstract

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety and $U$-operators. This conjecture is equivalent to the Kneser-Tits conjecture for simple, simply connected algebraic groups with Tits index $E^{78}_{8,2}$. We prove that a simple, simply connected algebraic group with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field of arbitrary characteristic, is $R$-trivial, in the sense of Manin, thereby proving the Kneser-Tits conjecture for such groups. The Tits-Weiss conjecture follows as a consequence.