Chapter

Lie groups and Lie algebras

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0002
Lie groups and Lie algebras

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Lie groups and Lie algebras are defined. SU(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators are defined and used to provide a basis for a vector space that leads to the Lie algebra. Structure constants are introduced and shown to provide the so-called adjoint representation. The Cartan metric tensor is defined and Cartan’s criterion for semisimplicity is described. After showing that SU(2) is compact SL(2,R) — the group whose elements are 2x2 real unimodular matrices — is introduced to give an example of a non-compact group. Biographical notes on Euler, Lie and Cartan are given.

Keywords: Lie groups; Lie algebras; infinitesimal generators; structure constants; adjoint representation; Cartan metric

Chapter.  3945 words. 

Subjects: Mathematical and Statistical Physics

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