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Automorphisms of Albert algebras and a conjecture of Tits and Weiss
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Automorphisms of Albert algebras and a conjecture of Tits and Weiss
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Journal Article
Automorphisms of Albert algebras and a conjecture of Tits and Weiss
2013
Overview
Let kk 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 kk which are pure first Tits constructions. The conjecture asserts that, for an Albert division algebra AA over a field kk, the structure group Str(A)Str(A) is generated by UU-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 kk-forms of E8E_8 with index E8,278E^{78}_{8,2} and anisotropic kernel a strict inner kk-form of E6E_6 correspond bijectively (via Moufang hexagons) to Albert division algebras over kk. The Kneser-Tits problem for a form of E8E_8 as above is equivalent to the Tits-Weiss conjecture (see Section 3). We provide a solution to the Kneser-Tits problem for kk-forms of E8E_8 corresponding to pure first Tits construction Albert division algebras. As an application, we prove that for the kk-group G=Aut(A), G(k)/R=1G=\\textbf {Aut}(A),~G(k)/R=1, where AA is an Albert division algebra over kk as above and RR stands for RR-equivalence in the sense of Manin.
Publisher
American Mathematical Society
Subject
