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A non-associative algebra (or distributive algebra) over a field ''K'' is a ''K''-vector space ''A'' equipped with a binary multiplication operation which is ''K''-bilinear ''A'' × ''A'' → ''A''. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidian space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. An algebra is ''unital'' or ''unitary'' if it has an identity element ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra. For example the octonions are unital, but Lie algebras never are. The nonassociative algebra structure of ''A'' may be studied by associating it with other associative algebras which are subalgebra of the full algebra of ''K''-endomorphisms of ''A'' as a ''K''-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing ''A''". More generally, some authors consider the concept of a non-associative algebra over a commutative ring ''R'': An ''R''-module equipped with an ''R''-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any ''R''-algebra), then it is naturally a -algebra, so some authors refer to non-associative -algebras as non-associative rings. == Algebras satisfying identities == Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities which simplify multiplication somewhat. These include the following identities. In the list, ''x'', ''y'' and ''z'' denote arbitrary elements of an algebra. * Associative: (''xy'')''z'' = ''x''(''yz''). * Commutative: ''xy'' = ''yx''. * Anticommutative:〔Schafer (1995) p.3〕 ''xy'' = −''yx''.〔This is always implied by the identity ''xx'' = 0 for all ''x'', and the converse holds for fields of characteristic other than two.〕 * Jacobi identity:〔〔Okubo (2005) p.12〕 (''xy'')''z'' + (''yz'')''x'' + (''zx'')''y'' = 0. * Jordan identity:〔Schafer (1995) p.91〕〔Okubo (2005) p.13〕 (''xy'')''x''2 = ''x''(''yx''2). * Power associative:〔Schafer (1995) p.30〕〔Okubo (2005) p.17〕〔 For all ''x'', any three nonnegative powers of ''x'' associate. That is if ''a'', ''b'' and ''c'' are nonnegative powers of ''x'', then ''a''(''bc'') = (''ab'')''c''. This is equivalent to saying that ''x''''m'' ''x''''n'' = ''x''''n+m'' for all non-negative integers ''m'' and ''n''. * Alternative:〔Schafer (1995) p.5〕〔Okubo (2005) p.18〕〔McCrimmon (2004) p.153〕 (''xx'')''y'' = ''x''(''xy'') and (''yx'')''x'' = ''y''(''xx''). * Flexible:〔Schafer (1995) p.28〕〔Okubo (2005) p.16〕 ''x''(''yx'') = (''xy'')''x''. * Elastic: Flexible and (''xy'')(''xx'') = ''x''(''y''(''xx'')), ''x''(''xx'')''y'' = (''xx'')(''xy''). These properties are related by # ''associative'' implies ''alternative'' implies ''power associative''; # ''associative'' implies ''Jordan identity'' implies ''power associative''; # Each of the properties ''associative'', ''commutative'', ''anticommutative'', ''Jordan identity'', and ''Jacobi identity'' individually imply ''flexible''.〔〔 # For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just . 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「Non-associative algebra」の詳細全文を読む スポンサード リンク
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