Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Abstract:

Let $k$ be a field of characteristic different from 2 and 3. 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. The conjecture asserts that, for an Albert division algebra $A$ over a field $k$, the structure group $Str(A)$ is generated by $U$-operators and scalar multiplications. The conjecture derives its importance from its connections with algebraic groups and Tits buildings, particularly with Moufang polygons. 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 Section 3). We provide a solution to the Kneser-Tits problem for $k$-forms of $E_8$ corresponding to pure first Tits construction Albert division algebras. As an application, we prove that for the $k$-group $G=\textbf {Aut}(A),~G(k)/R=1$, where $A$ is an Albert division algebra over $k$ as above and $R$ stands for $R$-equivalence in the sense of Manin.