R‐triviality of groups of type F4 arising from the first Tits construction

Abstract

Any group of type F4 is obtained as the automorphism group of an Albert algebra. We prove that such a group is R-trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras.

On $R$-triviality of $F_4$-II

“…In this paper, we prove that for an Albert algebra A over a field k of arbitrary characteristic, there exists an isotope A (v) such that the algebraic group Aut(A (v) ) is R-trivial. Since for first Tits construction Albert algebras all isotopes are isomorphic (see [17]), it follows that the algebraic group Aut(A) is R-trivial for first Tits construction Albert algebras, thereby improving the result proved in ( [1], [28]). The case of reduced Albert algebras which have no nonzero nilpotents and Albert division algebras in general is work in progress.…”

“…Let A = J(B, σ, u, µ) be an Albert algebra with S := (B, σ) + a division algebra and K = Z(B), a quadratic étale extension of k. Then any automorphism of A stabilizing S is of the form (a, b) → (pap −1 , pbq) for p ∈ U (B, σ) and q ∈ U (B, σ u ) with N B (p)N B (q) = 1. We have (1) , where…”

“…This work builds on the results from ( [28]). We recall at this stage that simple algebraic groups of type F 4 defined over a field k are precisely the full groups of automorphisms of Albert algebras defined over k. In papers ( [28], [1]), the authors proved (independently) that the automorphism group of an Albert algebra arising from the first Tits construction is R-trivial, the proof in ( [28]) being characteristic free.…”

On $R$-triviality of $F_4$

“…Let A = J(B, σ, u, µ) be an Albert algebra with S := (B, σ) + a division algebra and K = Z(B), a quadratic étale extension of k. Then any automorphism of A stabilizing S is of the form (a, b) → (pap −1 , pbq) for p ∈ U (B, σ) and q ∈ U (B, σ u ) with N B (p)N B (q) = 1. We have Aut(A, (B, σ) + ) ∼ = [U (B, σ) × U (B, σ u )] det /K (1) , where…”

“…We would like to mention that at the time this preprint was completed, S. Alsaody, V. Chernousov and A. Pianzola posted a preprint on the arXiv ( [1]), proving the R-triviality for groups of type F 4 that arise from the first Tits constructions, over fields of characteristic different from 2 and 3. Their proof relies on certain cohomological techniques.…”