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In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. (en) Die Darstellungstheorie von Algebren ist ein Teilgebiet der Mathematik, das sich mit der Darstellung von Algebren auf Vektorräumen beschäftigt. Auf diese Weise werden beliebige assoziative Algebren mittels Homomorphismen mit Algebren von Operatoren in Zusammenhang gebracht. Untersuchungsgegenstand sind die Struktur solcher Homomorphismen und deren Klassifikation. Die Darstellungstheorie einer Algebra ist zur Theorie ihrer Moduln äquivalent. Speziellere Darstellungstheorien behandeln Gruppen, Lie-Algebren oder C*-Algebren. Wir betrachten im Folgenden der Einfachheit halber Algebren mit Einselement 1. Hat man eine Algebra ohne Einselement, so adjungiere man eines. (de) 抽象代数学において，結合多元環の表現はその環の加群である．ここで結合多元環は（単位的とは限らない）環である．多元環が単位的でないとき，標準的な方法で単位的にでき（随伴関手のページを参照），得られる単位的環（単位元は恒等写像として作用する）の加群と多元環の表現の間に本質的な違いは存在しない． (ja) Em matemática, uma representação de uma álgebra é um módulo sobre a álgebra ou, equivalentemente, um homomorfismo de álgebras entre a álgebra e o de um espaço vetorial. (pt) |
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In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. (en) 抽象代数学において，結合多元環の表現はその環の加群である．ここで結合多元環は（単位的とは限らない）環である．多元環が単位的でないとき，標準的な方法で単位的にでき（随伴関手のページを参照），得られる単位的環（単位元は恒等写像として作用する）の加群と多元環の表現の間に本質的な違いは存在しない． (ja) Em matemática, uma representação de uma álgebra é um módulo sobre a álgebra ou, equivalentemente, um homomorfismo de álgebras entre a álgebra e o de um espaço vetorial. (pt) Die Darstellungstheorie von Algebren ist ein Teilgebiet der Mathematik, das sich mit der Darstellung von Algebren auf Vektorräumen beschäftigt. Auf diese Weise werden beliebige assoziative Algebren mittels Homomorphismen mit Algebren von Operatoren in Zusammenhang gebracht. Untersuchungsgegenstand sind die Struktur solcher Homomorphismen und deren Klassifikation. Die Darstellungstheorie einer Algebra ist zur Theorie ihrer Moduln äquivalent. Speziellere Darstellungstheorien behandeln Gruppen, Lie-Algebren oder C*-Algebren. (de) |
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