Totally decomposable symplectic and unitary involutions

TL;DR

This paper investigates the behavior of totally decomposable symplectic and unitary involutions on certain central simple algebras, revealing their isotropic or hyperbolic nature over field extensions, with new results in characteristic 2.

Contribution

It establishes new results on the isotropic or hyperbolic nature of these involutions over field extensions, especially in characteristic 2, extending previous work.

Findings

Involutions are either anisotropic or hyperbolic over any field extension.

The converse holds for algebras of 2-power degree.

New results are provided in characteristic 2.

Abstract

We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.