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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Separable algebra ： ウィキペディア英語版 Separable algebra In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. == Definition and First Properties == Let ''K'' be a field. An associative ''K''-algebra ''A'' is said to be separable if for every field extension $\backslash scriptstyle\; L/K$ the algebra $\backslash scriptstyle\; A\backslash otimes\_K\; L$ is semisimple. There is a classification theorem for separable algebras: separable algebras are the same as finite products of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field ''K''. If ''K'' is a perfect field --- for example a field of characteristic zero, or a finite field, or an algebraically closed field --- then every extension of ''K'' is separable. As a result, if ''K'' is a perfect field, separable algebras are the same as finite products of matrix algebras over division algebras whose centers are finite-dimensional field extensions of the field ''K''. In other words, if ''K'' is a perfect field, there is no difference between a separable algebra over ''K'' and a finite-dimensional semisimple algebra over ''K''. There are a several equivalent characterizations of separable algebras. First, an algebra ''A'' is separable if and only if there exists an element :$p\; =\; \backslash sum\_^n\; x\_i\; \backslash otimes\; y\_i$ in the ''enveloping algebra''〔Reiner (2003) p.101〕 $A^e\; =\; A\; \backslash otimes\_K\; A^$ such that : $\backslash sum\_^n\; x\_i\; y\_i\; =\; 1\_A$ and ''ap'' = ''pa'' for all ''a'' in ''A''. Such an element ''p'' is called a separability idempotent, since it satisfies $p^2\; =\; p$. A generalized theorem of Maschke shows these this characterization of separable algebras is equivalent to the definition given above. Second, an algebra ''A'' is separable if and only if it is projective when considered as a left module of $A^e$ in the usual way.〔Reiner (2003) p.102〕 Third, an algebra ''A'' is separable if and only if it is flat when considered as a right module of $A^e$ in the usual (but perhaps not quite standard) way. See Aguiar's note below for more details. Furthermore, a result of Eilenberg and Nakayama has that any separable algebra can be given the structure of a symmetric Frobenius algebra. Since the underlying vector space of a Frobenius algebra is isomorphic to its dual, any Frobenius algebra is necessarily finite dimensional, and so the same is true for separable algebras. A separable algebra is said to be strongly separable if there exists a separability idempotent that is symmetric, meaning :$e\; =\; \backslash sum\_^n\; x\_i\; \backslash otimes\; y\_i\; =\; \backslash sum\_^n\; y\_i\; \backslash otimes\; x\_i$ An algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a special Frobenius algebra. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Separable algebra」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.