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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Automorphisms of Albert algebras and a conjecture of Tits and Weiss II

Author: Maneesh Thakur
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 17C30; Secondary 20G15, 51E24
Published electronically: May 21, 2019
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Abstract: Let $ G$ be a simple, simply connected algebraic group with Tits index $ E_{8,2}^{78}$ or $ E_{7,1}^{78}$, defined over a field $ k$ of arbitrary characteristic. We prove that there exists a quadratic extension $ K$ of $ k$ such that $ G$ is $ R$-trivial over $ K$; i.e., for any extension $ F$ of $ K$, $ G(F)/R=\{1\}$, where $ G(F)/R$ denotes the group of $ R$-equivalence classes in $ G(F)$, in the sense of Manin. As a consequence, it follows that the variety $ G$ is retract $ K$-rational and that the Kneser-Tits conjecture holds for these groups over $ K$. Moreover, $ G(L)$ is projectively simple as an abstract group for any field extension $ L$ of $ K$. In their monograph, J. Tits and Richard Weiss conjectured that for an Albert division algebra $ A$ over a field $ k$, its structure group $ Str(A)$ is generated by scalar homotheties and its $ U$-operators. This is known to be equivalent to the Kneser-Tits conjecture for groups with Tits index $ E_{8,2}^{78}$. We settle this conjecture for Albert division algebras which are first constructions, in the affirmative. These results are obtained as corollaries to the main result, which shows that if $ A$ is an Albert division algebra which is a first construction and $ \Gamma $ its structure group, i.e., the algebraic group of the norm similarities of $ A$, then $ \Gamma (F)/R=\{1\}$ for any field extension $ F$ of $ k$; i.e., $ \Gamma $ is $ R$-trivial.

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Maneesh Thakur
Affiliation: Indian Statistical Institute, 7-S.J.S. Sansanwal Marg, New Delhi 110016, India

Keywords: Exceptional groups, algebraic groups, Albert algebras, structure group, Kneser--Tits conjecture
Received by editor(s): June 27, 2017
Received by editor(s) in revised form: September 11, 2018
Published electronically: May 21, 2019
Additional Notes: The author thanks Linus Kramer of the Mathematics Institute, University of Münster, for a visit in March 2016, when part of this paper was completed. The stay was supported by the Deutsche Forschungsgemeinschaft through SFB 878.
Article copyright: © Copyright 2019 American Mathematical Society