Central simple algebras with involution: a geometric approach

Abstract

Let k be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free PGL _n -varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of ${{\rm PGL}_n\rtimes\langle\tau\rangle}$, where τ is the automorphism of PGL _n given by τ (h) = (h ⁻¹)^transpose. Basic properties of an involution are described in terms of the action of ${{\rm PGL}_n\rtimes\langle\tau\rangle}$ on the associated variety, and in particular in terms of the stabilizer in general position for this action.