Automorphisms of Albert algebras and a conjecture of Tits and Weiss
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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.References
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Additional Information
- Maneesh Thakur
- Affiliation: Indian Statistical Institute, 7-S.J.S. Sansanwal Marg, New Delhi 110016, India
- MR Author ID: 368125
- Email: maneesh.thakur@gmail.com
- Received by editor(s): July 16, 2011
- Received by editor(s) in revised form: September 22, 2011
- Published electronically: November 28, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3041-3068
- MSC (2010): Primary 20G15; Secondary 17C30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05710-2
- MathSciNet review: 3034458