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AUTOMORPHISMS OF ALBERT ALGEBRAS AND A CONJECTURE OF TITS AND WEISS II.

Authors :
THAKUR, MANEESH
Source :
Transactions of the American Mathematical Society; 10/01/2019, Vol. 372 Issue 7, p4701-4728, 28p
Publication Year :
2019

Abstract

Let G be a simple, simply connected algebraic group with Tits index E<superscript>78</superscript> <subscript>8,2</subscript> or E<superscript>78</superscript> <subscript>7,1</subscript>, 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<superscript>78</superscript> <subscript>8,2</subscript>. 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 Γ its structure group, i.e., the algebraic group of the norm similarities of A, then Γ(F)/R = {1} for any field extension F of k; i.e., Γ is R-trivial. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00029947
Volume :
372
Issue :
7
Database :
Complementary Index
Journal :
Transactions of the American Mathematical Society
Publication Type :
Academic Journal
Accession number :
138764312
Full Text :
https://doi.org/10.1090/tran/7850