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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Automorphisms of Albert algebras and a conjecture of Tits and Weiss II
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by Maneesh Thakur PDF
Trans. Amer. Math. Soc. 372 (2019), 4701-4728 Request permission


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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Additional Information
  • Maneesh Thakur
  • Affiliation: Indian Statistical Institute, 7-S.J.S. Sansanwal Marg, New Delhi 110016, India
  • MR Author ID: 368125
  • Email:
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4701-4728
  • MSC (2010): Primary 17C30; Secondary 20G15, 51E24
  • DOI:
  • MathSciNet review: 4009439