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Automorphisms of Albert algebras and a conjecture of Tits and Weiss


Author: Maneesh Thakur
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
Published electronically: November 28, 2012
MathSciNet review: 3034458
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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={\bf 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.


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Additional Information

Maneesh Thakur
Affiliation: Indian Statistical Institute, 7-S.J.S. Sansanwal Marg, New Delhi 110016, India
Email: maneesh.thakur@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2012-05710-2
Keywords: Automorphisms, Albert algebras, structure group, inner structure group, Kneser-Tits
Received by editor(s): July 16, 2011
Received by editor(s) in revised form: September 22, 2011
Published electronically: November 28, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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