Peter–Weyl theorem (original) (raw)
Im mathematischen Teilgebiet der harmonischen Analyse verallgemeinert der Satz von Peter-Weyl, benannt nach Hermann Weyl und seinem Studenten Fritz Peter (1899–1949), die Fourierreihe für Funktionen auf beliebigen kompakten topologischen Gruppen.
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| dbo:abstract | Im mathematischen Teilgebiet der harmonischen Analyse verallgemeinert der Satz von Peter-Weyl, benannt nach Hermann Weyl und seinem Studenten Fritz Peter (1899–1949), die Fourierreihe für Funktionen auf beliebigen kompakten topologischen Gruppen. (de) El Teorema de Peter-Weyl es un resultado básico en la teoría del análisis armónico, aplicado a grupos topológicos que son compactos, pero no necesariamente . Hermann Weyl, junto con su estudiante Peter, lo probó en la configuración de un grupo compacto de Lie, G. El teorema generaliza los hechos significantes sobre la descomposición de la representación regular de un grupo finito, como fue descubierto por F.G. Frobenius e Issai Schur. Para establecer el Teorema, primero es necesaria la idea del Espacio de Hilbert sobre , ; esto es razonable puesto que la medida de Haar existe en . Llamando este espacio , el grupo tiene una representación unitaria en actuando por la derecha o por la izquierda. Esto implica una representación de vía Esta representación se descompone en la suma de por cada representación finita irreducible de G donde es la representación dual. Esto significa que hay una descripción de suma directa de con la indicación de todas las clases (hasta el isomorfismo) de representaciones unitarias irreducibles de . Esto implica inmediatamente la estructura de para las representaciones diestra o zurda de , que es la suma directa de cada ; tantas veces como su dimensión (siempre finita). (es) In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G. The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur. Let G be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the regular representation of G on L2(G) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L2(G). In the case that G is the group of unit complex numbers, this last result is simply a standard result from Fourier series. (en) Il teorema di Peter-Weyl è un risultato della teoria delle rappresentazioni che fornisce informazioni utili al calcolo delle rappresentazioni irriducibili di gruppi finiti (informazioni sul numero delle rappresentazioni irriducibili non equivalenti e sulla loro dimensione). Esso può anche essere usato per decomporre le rappresentazioni riducibili. In particolare afferma che le rappresentazioni irriducibili non equivalenti di un gruppo di ordine sono in numero finito uguale al numero delle classi di coniugio in cui il gruppo è suddiviso, e sono tali che l'insieme dei vettori di componenti al variare di che si ottengono al variare di da a e al variare di e da a (dimensione di ), formano una base ortonormale in . L'uso di questo teorema per i gruppi finiti viene ulteriormente semplificato introducendo la nozione di carattere, e ne esiste inoltre una generalizzazione per rappresentazioni di gruppi infiniti come ad esempio i gruppi di Lie. (it) 彼得-魏尔定理（英語：Peter–Weyl theorem）是调和分析和群表示论中的一组重要定理，于1927年由赫尔曼·魏尔和他的学生证明。该定理刻画了紧群不可约表示的完备性，可以视作有限群表示理论中弗罗贝尼乌斯定理的推广。定理分为三部分：第一部分指出，紧群的所有有限维不可约的，在上所有复值连续群函数构成、配备了的空间中稠密。第二部分指出，在任何一个可分希尔伯特空间上的酉表示都完全可约。第三部分断言，的所有有限维不可约酉表示的矩阵元构成了上平方可积的复值函数空间的一组标准正交基。 (zh) |
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| rdfs:comment | Im mathematischen Teilgebiet der harmonischen Analyse verallgemeinert der Satz von Peter-Weyl, benannt nach Hermann Weyl und seinem Studenten Fritz Peter (1899–1949), die Fourierreihe für Funktionen auf beliebigen kompakten topologischen Gruppen. (de) 彼得-魏尔定理（英語：Peter–Weyl theorem）是调和分析和群表示论中的一组重要定理，于1927年由赫尔曼·魏尔和他的学生证明。该定理刻画了紧群不可约表示的完备性，可以视作有限群表示理论中弗罗贝尼乌斯定理的推广。定理分为三部分：第一部分指出，紧群的所有有限维不可约的，在上所有复值连续群函数构成、配备了的空间中稠密。第二部分指出，在任何一个可分希尔伯特空间上的酉表示都完全可约。第三部分断言，的所有有限维不可约酉表示的矩阵元构成了上平方可积的复值函数空间的一组标准正交基。 (zh) El Teorema de Peter-Weyl es un resultado básico en la teoría del análisis armónico, aplicado a grupos topológicos que son compactos, pero no necesariamente . Hermann Weyl, junto con su estudiante Peter, lo probó en la configuración de un grupo compacto de Lie, G. El teorema generaliza los hechos significantes sobre la descomposición de la representación regular de un grupo finito, como fue descubierto por F.G. Frobenius e Issai Schur. (es) In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G. The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur. (en) Il teorema di Peter-Weyl è un risultato della teoria delle rappresentazioni che fornisce informazioni utili al calcolo delle rappresentazioni irriducibili di gruppi finiti (informazioni sul numero delle rappresentazioni irriducibili non equivalenti e sulla loro dimensione). Esso può anche essere usato per decomporre le rappresentazioni riducibili. L'uso di questo teorema per i gruppi finiti viene ulteriormente semplificato introducendo la nozione di carattere, e ne esiste inoltre una generalizzazione per rappresentazioni di gruppi infiniti come ad esempio i gruppi di Lie. (it) |
| rdfs:label | Satz von Peter-Weyl (de) Teorema de Peter-Weyl (es) Teorema di Peter-Weyl (it) 페터-바일 정리 (ko) Peter–Weyl theorem (en) Teorema de Peter-Weyl (pt) 彼得-魏尔定理 (zh) |
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