Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adjoint and coadjoint orbits of the Poincaré group

Richard Cushman, Wilberd van der Kallen

Abstract

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincaré group.

Adjoint and coadjoint orbits of the Poincaré group

Abstract

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincaré group.

Paper Structure

This paper contains 9 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Affine orthogonal group
  3. Classification of adjoint orbits
  4. Indecomposable distinguished types
  5. Adjoint orbits of the Poincaré group
  6. Classification of coadjoint orbits
  7. Coadjoint orbits of the Poincaré group
  8. Normal forms
  9. Acknowledgements

Key Result

Theorem 1

Every distinguished type is a sum of an indecomposable nilpotent distinguished type and a sum of indecomposable types. This decomposition is unique up to a reordering of the summands.

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 5
  • Example 6
  • Proposition 9
  • Lemma 10
  • Lemma 11
  • Remark 12
  • ...and 6 more