About: Representation of a Lie group

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In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play 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.

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rdf:type	Thing yago:WikicatLieGroups yago:Abstraction100002137 yago:Group100031264
rdfs:label	Representaciones de grupos de Lie (es) Représentation d'un groupe de Lie (fr) Rappresentazioni dei gruppi di Lie (it) リー群の表現 (ja) Representation of a Lie group (en) Представление группы Ли (ru) 李群表示 (zh)
rdfs:comment	Au croisement de la géométrie différentielle et de la théorie des représentations, la représentation des groupes de Lie est une approche de l'étude des groupes de Lie par représentation comme groupe d'automorphismes linéaires d'un espace vectoriel (voire comme groupe classique). Pour un groupe de Lie réel donné G, une représentation réelle ou complexe de G est la donnée d'un espace vectoriel réel ou complexe V et d'un morphisme de groupes de Lie de G dans GL(V), le groupe des automorphismes linéaires de V. (fr) In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play 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. (en) 数学や理論物理学では、リー群の表現の考え方は、連続対称性の研究で重要な役割を果たす。そのような表現は、対応する「無限小」リー代数の表現研究で使用する基本的なツールであることが良く知られている。物理学の文献では、リー群の表現とリー代数の表現との間の違いを強調しないこともある。 (ja) Si dice rappresentazione di un gruppo di Lie su uno spazio vettoriale un omomorfismo sotto il quale ogni elemento in è mappato in un elemento dello spazio degli operatori lineari invertibili agenti su e consistenti con le operazioni di gruppo. (it) Представление группы Ли — это линейное действие группы Ли на векторном пространстве или, что то же самое, гладкий гомоморфизм группы Ли в группу обратимых операторов на векторном пространстве. Играет важную роль в изучении непрерывной симметрии в математике и теоретической физике. Представления групп Ли изучены довольно хорошо, основным инструментом их изучения является использование соответствующих «инфинитезимальных» представлений алгебр Ли. (ru) 在数学和理论物理领域，李群表示（Representation of a Lie group）意指李群在向量空间上的线性作用。等价地说，群的表示是一个从该群到向量空间的可逆算子群的光滑同态。表示论在连续对称性的研究中扮演了重要的角色。关于这类表示的研究颇丰，其中一个基本的研究工具是使用对应的无穷小。 (zh) En matemáticas y física teórica, la idea de una representación de un grupo de Lie desempeña un papel importante en el estudio de la simetría continua. Mucho se sabe sobre tales representaciones, una herramienta básica en su estudio es el uso de las representaciones 'infinitesimales' correspondientes de las álgebras de Lie (de hecho en la literatura física la distinción a menudo se descuida). Si el homomorfismo es de hecho un monomorfismo, la representación se dice fiel. Una se define de la misma manera, excepto que G va en las matrices unitarias; las álgebras de Lie entonces mapean en . (es)
rdfs:seeAlso	Representation theory of SU(2)
foaf:depiction
dcterms:subject	Representation theory of Lie groups Lie groups
Wikipage page ID	292786 (xsd:integer)
Wikipage revision ID	1014260835 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Projective representation Pushforward (differential) Schrödinger equation Monomorphism Basis (linear algebra) Hydrogen atom Representation theory of SL2(R) Representation theory of the Lorentz group Character (mathematics) Vector space Lie algebra representation Lie group Wigner's classification (g,K)-module Representation theory of Lie groups Compact Lie group Mathematics Matrix exponential General linear group Clebsch–Gordan coefficients Representation theory of SU(2) Lorentz group MOS:NOTE MOS:PERSON Stone–von Neumann theorem Compact group Theoretical physics Symmetry Baker–Campbell–Hausdorff formula Wigner D-matrix Heisenberg group Irreducible representation Lie groups Harmonic analysis Simply connected space Group homomorphism Laplacian vector field Adjoint representation of a Lie group Tensor product Hydrogen-like atom Symmetry in quantum mechanics Homotopy group Maximal torus Weyl's theorem on complete reducibility Rotation group SO(3) Pontryagin duality Spherical harmonics Spin-½ Group representation Schur's lemma Unitary matrix Unitary representation Semisimple Lie algebra Poincaré group Weyl character formula Representation theory of Hopf algebras Theorem of the highest weight Half-integer spin Universal covering group Simply connected Lie correspondence Direct sum of representations List of Lie group topics Representation of Lie algebras Integer spin Space inversion symmetry Time reversal symmetry
Link from a Wikipage to an external page	http://www.math.sunysb.edu/~aknapp/books/beyond2.html%7Cedition= https://archive.org/details/quantumtheoryoff00stev
sameAs	Representation of a Lie group Representation of a Lie group Representation of a Lie group

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