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In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have * * 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 : 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〕 : : Both of these identities together imply that the associator is totally skew-symmetric. That is, : for any permutation σ. It follows that : for all ''x'' and ''y''. This is equivalent to the ''flexible identity''〔Schafer (1995) p.28〕 : 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: *right alternative identity: *flexible identity: 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
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