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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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