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Representation of a Lie group
In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras.〔 Chapter 2.〕
== Representations on a complex finite-dimensional vector space ==
Let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group ''G'' on a finite-dimensional complex vector space ''V'' is a smooth group homomorphism Ψ:''G''→Aut(''V'') from ''G'' to the automorphism group of ''V''.
For ''n''-dimensional ''V'', the automorphism group of ''V'' is identified with a subset of the complex square matrices of order ''n''. The automorphism group of ''V'' is given the structure of a smooth manifold using this identification. The condition that Ψ is smooth, in the definition above, means that Ψ is a smooth map from the smooth manifold ''G'' to the smooth manifold Aut(''V'').
If a basis for the complex vector space ''V'' is chosen, the representation can be expressed as a homomorphism into general linear group GL(''n'',C). This is known as a ''matrix representation''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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