Albert algebras and the Tits-Weiss conjecture

2019/11/09 by Maneesh Thakur, Thakur, Maneesh
Mathematics · #17C30 #20G15 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1911.04976

openalex publication_date 2019/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety and U-operators. This conjecture is equivalent to the Kneser-Tits conjecture for simple, simply connected algebraic groups with Tits index E⁷⁸_8,2. We prove that a simple, simply connected algebraic group with Tits index E_8,2⁷⁸ or E_7,1⁷⁸, defined over a field of arbitrary characteristic, is R-trivial, in the sense of Manin, thereby proving the Kneser-Tits conjecture for such groups. The Tits-Weiss conjecture follows as a consequence.

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