Some Group Theory

  • Syed Twareque Ali
  • Jean-Pierre Antoine
  • Jean-Pierre Gazeau
Part of the Theoretical and Mathematical Physics book series (TMP)


In this chapter, we provide the necessary group-theoretical background that will be used in later chapters. We start with the notion of homogeneous space of a locally compact group and its (quasi-) invariant measures. Then we turn to induced representations and the attending notion of system of covariance. This is illustrated by vector CS and the discrete series representations of SU(1,1). Next we describe briefly some aspects of harmonic analysis on a locally compact Abelian group. Finally we survey the basic facts concerning Lie groups and Lie algebras, their extensions and their contractions.

In this chapter, we introduce a few concepts from the theory of groups, Lie algebras, transformation spaces, and group representations, presenting them in a form and with notations adapted to the aims of this book. (A good source for more detailed information is, for example, [Bar77].)


Haar Measure Closed Subgroup Compact Abelian Group Coadjoint Orbit Invariant Subgroup 
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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Syed Twareque Ali
    • 1
  • Jean-Pierre Antoine
    • 2
  • Jean-Pierre Gazeau
    • 3
    • 4
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Institut de Recherche en Mathématique et Physique (IRMP)Université Catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Astroparticules et Cosmologie (APC, UMR 7164)Université Paris DiderotSorbonne Paris Cité ParisFrance
  4. 4.Centro Brasileiro de Pesquisas Fisicas (CBPF)Rio de JaneiroBrasil

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