Orders of Quaternion Algebras with Involution

TL;DR

This paper introduces maximal orders in quaternion algebras with orthogonal involution, providing a classification over local fields and partial results over algebraic number fields, advancing understanding of their algebraic structure.

Contribution

It presents the first classification of maximal orders with involution over local fields and offers partial classification over number fields, filling a gap in algebraic theory.

Findings

Classification over local fields achieved

Partial classification over algebraic number fields provided

Framework for understanding orders with involution established

Abstract

We introduce the notion of maximal orders over quaternion algebras with orthogonal involution and give a classification over local fields, and a partial classification over algebraic number fields.