Isotopy and invariants of Albert algebras

Abstract

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras $J(B,\sigma ,u,\mu )$ and $J(B,\tau ,v,\nu )$ have same $f_3$ and $g_3$ invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with $f_3(J^{\prime })=0$ , $f_5(J^{\prime })=0$ and $g_3(J^{\prime })=g_3(J)$ . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions.