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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 Superalgebra ： ウィキペディア英語版 Superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.〔Kac, Martinez & Zelmanov (2001), (p. 3 ).〕 That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. ==Formal definition== Let ''K'' be a fixed commutative ring. In most applications, ''K'' is a field such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :$A\; =\; A\_0\backslash oplus\; A\_1$ together with a bilinear multiplication ''A'' × ''A'' → ''A'' such that :$A\_iA\_j\; \backslash sube\; A\_$ where the subscripts are read modulo 2. A superring, or Z2-graded ring, is a superalgebra over the ring of integers Z. The elements of ''A''''i'' are said to be homogeneous. The parity of a homogeneous element ''x'', denoted by |''x''|, is 0 or 1 according to whether it is in ''A''0 or ''A''1. Elements of parity 0 are said to be even and those of parity 1 to be odd. If ''x'' and ''y'' are both homogeneous then so is the product ''xy'' and $|xy|\; =\; |x|\; +\; |y|.$ An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital. A commutative superalgebra (or supercommutative algebra) is one which satisfies a graded version of commutativity. Specifically, ''A'' is commutative if :$yx\; =\; (-1)^xy\backslash ,$ for all homogeneous elements ''x'' and ''y'' of ''A''. 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「Superalgebra」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.