Vahlen groups defined over commutative rings

TL;DR

This paper generalizes the concept of Vahlen groups, originally defined over real or complex Clifford algebras, to be over arbitrary commutative rings with quadratic forms, establishing their group structure and relation to pin and spin groups.

Contribution

It extends the definition of Vahlen groups to commutative rings with quadratic forms and proves their group properties and connections to Clifford-related groups.

Findings

Vahlen groups are groups over commutative rings with quadratic forms.

They are identified as pin, spin, or related groups within the Clifford group.

The generalized definitions are equivalent to standard ones under mild conditions.

Abstract

Elements of a Vahlen group are $2 \times 2$ matrices with entries in a Clifford algebra satisfying some conditions. Traditionally they have come in both ordinary and paravector type and have been defined (over Clifford algebras) over the real or complex numbers. We extend the definition of both types to be over a commutative ring with an arbitrary quadratic form. We show that they are indeed groups and identify in each case the group as the pin group, spin group, or another subgroup of the Clifford group. Under some mild conditions, for both types we show the equivalence of our definition with a suitably generalised version of the two standard definitions.