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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 Alternative algebra ： ウィキペディア英語版 Alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *$x(xy)\; =\; (xx)y$ *$(yx)x\; =\; y(xx)$ for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The sedenions, on the other hand, are not alternative. ==The associator== Alternative algebras are so named because they are precisely the algebras for which the associator is alternating. The associator is a trilinear map given by :$()\; =\; (xy)z\; -\; x(yz)$ By definition a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to〔Schafer (1995) p.27〕 :$()\; =\; 0$ :$()\; =\; 0.$ Both of these identities together imply that the associator is totally skew-symmetric. That is, :$(x\_,\; x\_)\; =\; \backslash sgn(\backslash sigma)()$ for any permutation σ. It follows that :$()\; =\; 0$ for all ''x'' and ''y''. This is equivalent to the ''flexible identity''〔Schafer (1995) p.28〕 :$(xy)x\; =\; x(yx).$ The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of: *left alternative identity: $x(xy)\; =\; (xx)y$ *right alternative identity: $(yx)x\; =\; y(xx)$ *flexible identity: $(xy)x\; =\; x(yx).$ is alternative and therefore satisfies all three identities. An alternating associator is always totally skew-symmetric. The converse holds so long as the characteristic of the base field is not 2. 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「Alternative 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.