Lie Groups and Lie Algebras

  • Wolfgang Ludwig
  • Claus Falter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 64)


Whereas discrete groups mainly describe the symmetries of regular geometric structures (crystals), continuous groups are essential in discussing the properties of particles, fields (atoms and all the more elementary particles) and conservation laws. We restrict the investigation here to Lie groups and the Lie algebras connected with them. First we discuss the fundamental notions and relations for these groups, which are the generators, the (unitary) REPs, the invariants, connected spaces, covering and compact groups, simple and semisimple groups, etc. Most of the ideas are illustrated with the Y U(n) groups. The central notions like symmetry projection operators, Clebsch-Gordan decomposition and the Wigner-Eckart theorem known from finite groups are generalized to continuous ones. For the investigation of semisimple groups and their IRs the knowledge of their weight and root systems is extremely useful.


Structure Constant Maximal Weight Young Diagram Casimir Operator Weight System 
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  1. 11.1
    B.G. Wybourne: Classical Groups for Physicists (Wiley, New York 1974)MATHGoogle Scholar
  2. 11.2
    G. Racah: Sulla caratterizzazione delle rappresentazione irriducibili dei gruppi semisimplici di Lie. Lincei Rend. Sci. Fis. Mat. Nat. 8, 108 (1950)MATHMathSciNetGoogle Scholar
  3. 11.3
    E.G. Beltrametti, A. Blasi: On the number of Casimir operators associated with any Lie group. Phys. Lett. 20, 62 (1966)CrossRefMATHADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Wolfgang Ludwig
    • 1
  • Claus Falter
    • 1
  1. 1.Institut für Theoretische PhysikWestfälische Wilhelms-UniversitätMünsterGermany

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