Adjoint representation (original) (raw)
In der Mathematik spielen die adjungierten Darstellungen von Lie-Gruppen und Lie-Algebren eine wichtige Rolle in Differentialgeometrie, Darstellungstheorie und Mathematischer Physik.
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| dbo:abstract | 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: . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. (en) 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) 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 . Per ogni gruppo di Lie, questa rappresentazione naturale si ottiene linearizzando (cioè prendendo il differenziale) l'azione del gruppo su se stesso per coniugazione. La rappresentazione aggiunta può essere definita per gruppi algebrici lineari su campi arbitrari. (it) 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) |
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| 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: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) |
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