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

Article Type

Research Article

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Transactions of the American Mathematical Society


Let G be a simple, simply connected algebraic group with Tits index E8782 or E7781, 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 E8782. 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.

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