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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

(p.8) 2 Lie groups and Lie algebras
Lie Groups and Lie Algebras

Adam M. Bincer

Oxford University Press

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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