Clifford Groups of Arbitrary Quadratic Modules over Commutative Rings

TL;DR

This paper generalizes the theory of Clifford algebras and groups to quadratic modules over arbitrary commutative rings, exploring subalgebras and relations with orthogonal groups.

Contribution

It extends classical Clifford algebra and group results to quadratic modules over arbitrary commutative rings, including degenerate cases.

Findings

Identified important subalgebras of Clifford algebras under certain conditions

Generalized relations between Clifford groups and orthogonal groups

Applicable to quadratic modules over arbitrary commutative rings

Abstract

We consider the Clifford algebra and the Clifford group associated with any quadratic module, degenerate or not, over an arbitrary commutative ring with 1. We determine some of the important subalgebras of the Clifford algebra under some conditions, and generalize some of the classical relations between the Clifford group and the orthogonal group of the quadratic module.