Pfister involutions

Abstract

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.

Orthogonal Pfister involutions in characteristic two

“…The implication (1) ⇒ (3), and the equivalence (2) ⇔ (3), follows from the Pfister Factor Conjecture and the non-hyperbolic splitting result of [8]. The converse implication, (2) or (3) ⇒ (1), is still open in general.…”

“…Assuming that the characteristic of F is different from 2, in [2] it is asked whether the following are equivalent:…”

“…The theory of symmetric bilinear forms in characteristic 2 has several features that mean we must be slightly more careful in our formulation of the analogous question to that posed in [2]. Over fields of characteristic different from 2, all 2-dimensional isotropic symmetric bilinear forms are isometric to the hyperbolic plane.…”

“…We shall also consider potential analogues of the other implications from the quesion asked in [2]. That a split totally decomposable F -algebra with orthogonal involution is adjoint to a bilinear Pfister form in charactersitic 2, an analogue of (1) ⇒ (2), can be shown with relatively simple arguments (see (6.7)).…”

Totally decomposable symplectic and unitary involutions

“…Over fields of characteristic different from two, the adjoint involution of a Pfister form is a totally decomposable involution. Pfister forms are a central part of the algebraic theory of quadratic forms, and in [2] it was asked whether totally decomposable involutions share certain characterising properties of Pfister forms. In particular, whether totally decomposable involutions are exactly those involutions on algebras of two-power degree that are either anisotropic or hyperbolic after extending scalars.…”