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In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory,. Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.

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  • Frobenius algebra (en)
  • Algèbre de Frobenius (fr)
  • フロベニウス多元環 (ja)
  • 프로베니우스 대수 (ko)
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  • In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory,. Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory. (en)
  • Une algèbre associative unifère de dimension finie A définie sur un corps k est appelée une algèbre de Frobenius si A est munie d'une forme bilinéaire non dégénérée σ:A × A → k qui satisfait l'équation suivante: σ(a·b,c)=σ(a,b·c). Cette forme bilinéaire est appelée la forme de Frobenius de cette algèbre. (fr)
  • フロベニウス多元環(フロベニウスたげんかん、英: Frobenius algebra)、あるいはフロベニウス代数とは、数学の表現論や加群論において有限次元な単位的結合多元環のうち、良い双対理論を与える特別な双線型形式を持つものをいう。 フロベニウス多元環は1930年代に と によって有限群のモジュラー表現の一般化として研究され始め、Frobenius にちなんで名づけられた。中山は および特に において豊かな双対理論を初めて発見した。デュドネはこれを用いて においてフロベニウス多元環を特徴づけ、フロベニウス多元環のこの性質を perfect duality と呼んだ。フロベニウス多元環は準フロベニウス環(右正則表現が移入的なネーター環)へと一般化された。最近では、フロベニウス多元環への関心は、位相的場の理論との関連からも高まっている。 体上の有限次元多元環に対しては以下のようなクラスの階層がある。 自己入射多元環 ⊃ フロベニウス多元環 ⊃ 対称多元環 ⊃ 半単純多元環 ⊃ 単純多元環 ⊃ 可除多元環 (ja)
  • 추상대수학에서 프로베니우스 대수(영어: Frobenius algebra)는 호환되는 내적이 주어진 유한 차원 단위 결합 대수이다. (ko)
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