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Lie Groups and Lie AlgebrasA Physicist's Perspective$
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Adam M. Bincer

Print publication date: 2012

Print ISBN-13: 9780199662920

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199662920.001.0001

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Lie groups and Lie algebras

Lie groups and Lie algebras

Chapter:
(p.8) 2 Lie groups and Lie algebras
Source:
Lie Groups and Lie Algebras
Author(s):

Adam M. Bincer

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199662920.003.0002

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

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