Abstract

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Highlights

  • By a metric vector space we mean a finite dimensional vector space V that is endowed with a quadratic form Q

  • We follow the exposition by Schroder [50] and provide in Sect. 2 basic facts about a metric vector space (V, Q) and its weak orthogonal group O (V, Q), which in most cases is generated by reflections

  • Math representation of the Lipschitz group Lip×(V, Q), which provides a surjective homomorphism onto the weak orthogonal group O (V, Q)

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Summary

Introduction

By a metric vector space we mean a finite dimensional vector space V (over a field F of arbitrary characteristic) that is endowed with a quadratic form Q. The description of orthogonal transformations of a metric vector space (V , Q) in terms of its associated Clifford algebra Cl(V , Q) has a long history. We follow the exposition by Schroder [50] and provide in Sect. 2 basic facts about a metric vector space (V , Q) and its weak orthogonal group O (V , Q), which in most cases is generated by reflections. 3, we collect from various sources those results about the Clifford algebra Cl(V , Q) which are needed later on. We recall from there the Lipschitz monoid Lip(V , Q) and the twisted adjoint

Results
Metric Vector Spaces
Clifford Algebras
Lipschitz Groups
Projective Metric Geometry
A Comparison of Clifford Algebras
Future Research
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