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| {{Lie groups |Representation}}
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| 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' [[representation of Lie algebras|representations of Lie algebras]]. The physics literature sometimes passes over the distinction between Lie group representations and Lie algebra representations.
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| == Representations on a complex finite-dimensional vector space ==
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| Let us first discuss representations acting on finite-dimensional complex vector spaces. A [[group representation|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''.
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| For ''n''-dimensional ''V'', the automorphism group of ''V'' is identified with a subset of 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'').
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| 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''.
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| == Representations on a finite-dimensional vector space over an arbitrary field ==
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| A [[group representation|representation]] of a [[Lie group]] ''G'' on a [[vector space]] ''V'' (over a [[field (mathematics)|field]] ''K'') is a [[smooth function|smooth]] (i.e. respecting the differential structure) [[group homomorphism]] ''G''→Aut(''V'') from ''G'' to the [[automorphism group]] of ''V''. If a basis for the vector space ''V'' is chosen, the representation can be expressed as a homomorphism into [[general linear group|GL(''n'',''K'')]]. This is known as a ''matrix representation''.
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| Two representations of ''G'' on vector spaces ''V'', ''W'' are ''equivalent'' if they have the
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| same matrix representations with respect to some choices of bases
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| for ''V'' and ''W''.
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| On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to [[endomorphism|End(''V'')]] preserving the [[Lie_algebra#Definition_and_first_properties|Lie bracket]] [ , ]. See [[representation of Lie algebras]] for the Lie algebra theory.
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| If the homomorphism is in fact a [[monomorphism]], the representation is said to be ''faithful''.
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| A [[unitary representation]] is defined in the same way, except that G maps to [[unitary matrix|unitary matrices]]; the Lie algebra will then map to [[skew-hermitian]] matrices.
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| If ''G'' is a compact Lie group, every finite-dimensional representation is equivalent to
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| a unitary one.
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| == Representations on Hilbert spaces ==
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| A [[group representation|representation]] of a [[Lie group]] ''G'' on a complex [[Hilbert space]] ''V'' is a [[group homomorphism]] Ψ:''G'' → B(''V'') from ''G'' to B(''V''), the group of bounded linear operators of ''V'' which have a bounded inverse, such that the map ''G''×''V'' → ''V'' given by (''g'',''v'') → Ψ(''g'')''v'' is continuous.
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| This definition can handle representations on '''infinite-dimensional''' Hilbert spaces. Such representations can be found in e.g. quantum mechanics, but also in Fourier analysis as shown in the following example.
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| Let ''G''='''R''', and let the complex Hilbert space ''V'' be ''L''<sup>2</sup>('''R'''). We define the representation Ψ:'''R''' → B(''L''<sup>2</sup>('''R''')) by Ψ(''r''){''f''(''x'')} → ''f''(''r''<sup>-1</sup>''x'').
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| See also [[Wigner's classification]] for representations of the [[Poincaré group]].
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| == Classification ==
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| If G is a [[Semisimple Lie group|semisimple]] group, its finite-dimensional representations can be decomposed as [[direct sum of representations|direct sums]] of [[irreducible representation]]s. The irreducibles are indexed by highest [[weight (representation theory)|weight]]; the allowable (''dominant'') highest weights satisfy a suitable positivity condition. In particular, there exists a set of ''fundamental weights'', indexed by the vertices of the [[Dynkin diagram]] of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights. The characters of the irreducible representations are given by the [[Weyl character formula]].
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| If G is a commutative [[Lie group]], then its irreducible representations are simply the continuous [[Character (mathematics)|character]]s of G: see [[Pontryagin duality]] for this case.
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| A quotient representation is a [[quotient module]] of the [[group ring]].
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| == Formulaic examples ==
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| Let '''F'''<sub>''q''</sub> be a finite field of order ''q'' and characteristic ''p''. Let ''G'' be a finite group of Lie type, that is, ''G'' is the '''F'''<sub>''q''</sub>-rational points of a connected reductive group ''G'' defined over '''F'''<sub>''q''</sub>. For example, if ''n'' is a positive integer GL(''n'', '''F'''<sub>''q''</sub>) and SL(n, '''F'''<sub>''q''</sub>) are finite groups of Lie type. Let <math>J = \left [ \begin{smallmatrix}0 & I_n \\ -I_n & 0\end{smallmatrix} \right ]</math>, where ''I''<sub>n</sub> is the ''n''×''n'' identity matrix. Let
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| : <math>Sp_2(\mathbb{F}_q) = \left \{ g \in GL_{2n}(\mathbb{F}_q) | ^tgJg = J \right \}.</math>
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| Then Sp(2,'''F'''<sub>''q''</sub>) is a symplectic group of rank ''n'' and is a finite group of Lie type. For ''G'' = GL(''n'', '''F'''<sub>''q''</sub>) or SL(''n'', '''F'''<sub>''q''</sub>) (and some other examples), the ''[[standard Borel subgroup]]'' ''B'' of ''G'' is the subgroup of ''G'' consisting of the upper triangular elements in ''G''. A ''[[standard parabolic subgroup]]'' of ''G'' is a subgroup of ''G'' which contains the standard Borel subgroup ''B''. If ''P'' is a standard parabolic subgroup of GL(''n'', '''F'''<sub>''q''</sub>), then there exists a partition (''n''<sub>1</sub>, …, ''n''<sub>r</sub>) of ''n'' (a set of positive integers <math>n_j\,\!</math> such that <math>n_1 + \ldots + n_r = n\,\!</math>) such that <math>P = P_{(n_1,\ldots,n_r)} = M \times N</math>, where <math>M \simeq GL_{n_1}(\mathbb{F}_q) \times \ldots \times GL_{n_r}(\mathbb{F}_q)</math> has the form
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| : <math>M = \left \{\left.\begin{pmatrix}A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & A_r\end{pmatrix}\right|A_j \in GL_{n_j}(\mathbb{F}_q), 1 \le j \le r \right \},</math> | |
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| and
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| : <math>N=\left \{\begin{pmatrix}I_{n_1} & * & \cdots & * \\ 0 & I_{n_2} & \cdots & * \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & I_{n_r}\end{pmatrix}\right\},</math>
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| where <math>*\,\!</math> denotes arbitrary entries in <math>\mathbb{F}_q</math>.
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| <!--''This section is still in progress. It should be finished soon.''[[User:Vermin1302|Vermi]] 06:22, 26 October 2005 (UTC) -->
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| ==See also==
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| *[[Representation theory of the Lorentz group]]
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| *[[Representation theory of Hopf algebras]]
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| *[[Adjoint representation of a Lie group]]
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| *[[List of Lie group topics]]
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| * [[Symmetry in quantum mechanics]]
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| == References ==
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| * {{Fulton-Harris}}
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| * {{Citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction |publisher=Springer|year=2003|isbn=0-387-40122-9}}.
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| * {{Citation|last=Knapp|first=Anthony W.|authorlink=Anthony Knapp|title=Lie Groups Beyond an Introduction|url=http://www.math.sunysb.edu/~aknapp/books/beyond2.html|edition= 2nd|series=Progress in Mathematics|volume=140|publisher=Birkhäuser|place= Boston|year= 2002}}.
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| * {{Citation|last=Rossmann|first= Wulf |title=Lie Groups: An Introduction Through Linear Groups|series= Oxford Graduate Texts in Mathematics|publisher= Oxford University Press|isbn= 978-0-19-859683-7|year=2001}}. The 2003 reprint corrects several typographical mistakes.
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| [[Category:Lie groups]]
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| [[Category:Representation theory of Lie groups]]
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Greetings. The author's identify is Annabell Traxler. One of her most loved hobbies is fishing and now she has time to consider on new issues. Her position is a financial debt collector and her income has been truly fulfilling. Delaware is the put she enjoys most. Examine out the most up-to-date news on her web-site: http://checkmywheels.co.uk/members/tiffaspivakov/activity/25166/
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