2017/04/25 by Maneesh Thakur, Thakur, Maneesh
Mathematics · #17A40 #17C30 #20G15 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1704.07811
openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main aim of this paper is to prove R-triviality for simple, simply connected algebraic groups with Tits index E8,278 or E7,178, defined over a field k of arbitrary characteristic. Let G be such a group. 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 (see \citeM). 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 (\citeTW) 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 E8,278. We settle this conjecture for Albert division algebras which are first constructions, in 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.