Algebraic Groups Constructed from Orders of Quaternion Algebras

TL;DR

This paper explores how spin groups of certain quadratic forms can be represented using quaternion algebra matrices, and investigates their arithmetic subgroups through orders of these algebras, linking algebraic properties to group isomorphisms.

Contribution

It provides a novel representation of spin groups via quaternion algebra matrices and analyzes their arithmetic subgroups through orders, establishing conditions for group isomorphisms.

Findings

Spin groups of non-definite, quinary quadratic forms can be represented as 2x2 matrices over quaternion algebras.

Maximal arithmetic subgroups can be constructed from orders of quaternion algebras.

Conditions for group isomorphism and conjugacy are characterized by algebraic properties of the underlying rings.

Abstract

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study maximal arithmetic subgroups of such groups, and show that examples can be produced by studying orders of the quaternion algebra. In both cases, we relate the algebraic properties of the underlying rings to sufficient and necessary conditions for the groups to be isomorphic and/or conjugate to one another.