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Projective metric geometry and Clifford algebras

Hans Havlicek

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.

Projective metric geometry and Clifford algebras

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.

Paper Structure

This paper contains 7 sections, 7 theorems, 42 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Metric vector spaces
  3. Clifford algebras
  4. Lipschitz groups
  5. Projective metric geometry
  6. A comparison of Clifford algebras
  7. Future research

Key Result

Lemma 4.2

Let $({\bm V},Q)$ be a metric vector space. Then the kernel of the twisted adjoint representation $\xi$ of the Lipschitz group $\mathop{\mathrm{Lip}}\nolimits^\times({\bm V},Q)$ satisfies the following properties.

Theorems & Definitions (20)

  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 5.1
  • Remark 5.2
  • Remark 5.3
  • Theorem 5.4
  • proof
  • Theorem 5.5
  • proof
  • ...and 10 more