About: Adjoint representation

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In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .

Attributes	Values
rdf:type	Thing
rdfs:label	Adjungierte Darstellung (de) Adjoint representation (en) Rappresentazione aggiunta (it) Représentation adjointe (fr) 随伴表現 (ja) 딸림표현 (ko) Representação adjunta (grupo de Lie) (pt) Присоединённое представление группы Ли (ru) Приєднане представлення групи Лі (uk) 伴随表示 (zh)
rdfs:comment	In der Mathematik spielen die adjungierten Darstellungen von Lie-Gruppen und Lie-Algebren eine wichtige Rolle in Differentialgeometrie, Darstellungstheorie und Mathematischer Physik. (de) En mathématiques, il existe deux notions de représentations adjointes : * la représentation adjointe d'un groupe de Lie sur son algèbre de Lie, * la représentation adjointe d'une algèbre de Lie sur elle-même. Alors que la première est une représentation de groupe, la seconde est une représentation d'algèbre. (fr) 리 군론에서 딸림표현(-表現, 영어: adjoint representation)은 어떤 리 군이 스스로의 리 대수 위에 가지는 표준적인 표현이다. (ko) リー群のリー環上への随伴表現（ずいはんひょうげん、英: adjoint representation）とは、リー群の元をリー環のある種の線型変換として表したものをいう。 (ja) Em matemática, a representação adjunta (ou ação adjunta) de um grupo de Lie G é uma forma de representar os elementos do grupo como transformações lineares do grupo de álgebra de Lie, considerado como um espaço vetorial. Por exemplo, no caso em que G é o grupo de Lie de matrizes inversíveis de tamanho n, GL(n), a álgebra de Lie é o espaço vetorial de todas (não necessariamente inversível) matrizes n-por-n. Portanto, neste caso, a representação adjunta é o espaço vetorial de matrizes n-por-n, e qualquer elemento g em GL(n) que atua como uma transformação linear deste espaço vetorial dada pela conjugação: . (pt) Присоединённое представление группы Ли — линейное представление группы Ли на своей алгебре Ли.Обычно обозначается . (ru) У теорії груп Лі приєднаним представленням групи Лі G називається представлення елементів групи, як лінійних відображень на відповідній алгебрі Лі. Дане представлення є гомоморфізмом груп Лі. Його диференціал є представленням алгебри Лі, що називається приєднаним представленням алгебри Лі. (uk) 在數學中，一個李群 G 的伴隨表示（adjoint representation）或伴隨作用（adjoint action）是 G 在它自身的李代數上的自然表示。這個表示是群 G 在自身上的共軛作用的線性化形式。 (zh) In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: . (en) In matematica, la rappresentazione aggiunta (o azione aggiunta) di un gruppo di Lie è un modo di rappresentare gli elementi del gruppo come trasformazioni lineari dell'algebra di Lie del gruppo, considerata come uno spazio vettoriale. Ad esempio, dato il gruppo , il gruppo di Lie di matrici invertibili reali n per n, allora la rappresentazione aggiunta è l'omomorfismo di gruppo che manda una matrice n-per-n invertibile a un endomorfismo dello spazio vettoriale di tutte le trasformazioni lineari di definito da . (it)
rdfs:seeAlso	Representation theory
dcterms:subject	Representation theory of Lie groups Lie groups
Wikipage page ID	302188 (xsd:integer)
Wikipage revision ID	1101102744 (xsd:integer)
Link from a Wikipage to another Wikipage	Representation theory Derivative of the exponential map Determinant Algebraic group Vector field Vector space Jacobi identity Lie's third theorem Lie derivative Lie group Lie group–Lie algebra correspondence Representation theory of Lie groups Mathematics Matrix multiplication General Leibniz rule SL2(R) Endomorphism Connected space Lie algebra Lie algebra extension Identity component Structure constants Symplectic manifold Tangent space Center (group theory) Trace (linear algebra) Linear algebraic group Linear map Alexandre Kirillov Lie groups Field (mathematics) Flow (mathematics) Center (algebra) Differential of a function Faithful representation Isotropy representation Conjugation (group theory) First isomorphism theorem Quadratic form Group action (mathematics) Invertible matrix Abelian group Chain rule Kernel (group theory) Coadjoint representation Maximal torus Weight (representation theory) Differential algebra Automorphism group Manifold Group representation Inner automorphism Orbit (group theory) Kirillov character formula Lie bracket of vector fields Root system Exponential map (Lie theory) Image (mathematics) Linear transformation Basis vectors Centralizer Lie algebra automorphism Matrix Lie group Representation of a Lie algebra Contragredient representation Semisimple group Nilpotent Lie group
sameAs	Adjoint representation Adjoint representation Adjoint representation Adjoint representation Adjoint representation Adjoint representation Adjoint representation

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