JournalscmhVol. 74 , No. 2DOI 10.1007/s000140050090

Isotopy and invariants of Albert algebras

  • Tata Institute of Fundamental Research, Colaba, Mumbai, India
Isotopy and invariants of Albert algebras cover

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,σ,u,μ)J(B,\sigma,u,\mu) and J(B,τ,v,ν)J(B,\tau,v,\nu) have same f3f_3 and g3g_3 invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with f3(J)=0f_3(J')=0 , f5(J)=0f_5(J')=0 and g3(J)=g3(J)g_3(J')=g_3(J) . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions.