Lie Groups and Lie Algebras

Abstract

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.KeywordsStructure ConstantMaximal WeightYoung DiagramCasimir OperatorWeight SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.