Abstract
AbstractWe prove that the structure group of any Albert algebra over an arbitrary field isR-trivial. This implies the Tits–Weiss conjecture for Albert algebras and the Kneser–Tits conjecture for isotropic groups of type$\mathrm {E}_{7,1}^{78}, \mathrm {E}_{8,2}^{78}$. As a further corollary, we show that some standard conjectures on the groups ofR-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type$^1\mathrm {E}_6$.
Highlights
The primary aim of this article is to prove the long standing Tits–Weiss conjecture on U-operators in Albert algebras and the Kneser–Tits conjecture for algebraic groups of type E778,1 and E788,2.The Tits–Weiss conjecture asserts that the structure group Str( A) of an arbitrary Albert algebra A is generated by the inner structure group, formed by the so-called U-operators, and the central homotheties
This problem was raised by Tits and Weiss in their 2002 book [26], where they studied spherical buildings and the corresponding generalised polygons attached to isotropic groups of relative rank 2
Rank 1 groups are useful in studying isotropic groups of exceptional types, where algebraic groups and their associated root subgroups are typically parametrised by a nonassociative structure and, as emphasised in [6], a rich interplay emerges between rank 1 groups, nonassociative algebras and linear algebraic groups
Summary
The primary aim of this article is to prove the long standing Tits–Weiss conjecture on U-operators in Albert algebras and the Kneser–Tits conjecture for algebraic groups of type E778,1 and E788,2. We would like to mention that it follows directly from our main theorem that two standard conjectures hold for simple connected strongly inner forms of type E6: the abelian nature of the group of R-equivalence classes and the existence of transfers for the functor of R-equivalence classes. For these groups, the norm principle holds as well. Preliminaries For later use we record some facts about Albert algebras and algebraic groups
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