Automorphisms of Albert algebras and a conjecture of Tits and Weiss

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 [31]). 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 = 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 ([10]).

$E_8$, the most exceptional group

“…One of the remaining open cases for E 8 was recently settled in [127], and another was settled in some special cases in [153].…”

R‐triviality of groups of type F4 arising from the first Tits construction

“…Proof Let be the corresponding Albert algebra. By [23, 24], the structure group is ‐trivial. Hence the assertion follows from the above corollary.…”

The cyclicity problem for Albert algebras

“…Albert division algebras that are first Tits constructions contain cyclic cubic subfields (see §2.c-(iii)). This fact was exploited in ( [21] and [22]) to prove the Tits-Weiss conjecture for the groups of type E 8 that arise from Albert division algebras that are first Tits constructions. We refer to ([3] and [7]) and references therein for further connections to Lie theory.…”