| Non-associative Algebra |
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| CATEGORIES ABOUT NON-ASSOCIATIVE ALGEBRA | |
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The associative law is the rule of ordinary algebra allowing one to dispense with parentheses in dealing with addition, or multiplication: it makes good sense to write down 3 + 5 + 7, and there is no requirement to specify (3 + 5) + 7, or 3 + (5 + 7). In Group Theory and many other branches of algebra the associative law is also assumed. A reason lying behind this is that Composition Of Functions is associative; so that abstraction away from function composition may still permit the associative law to be assumed. The study of non-associative structures therefore arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra that has grown very large is that of Lie Algebra s. There the associative law is replaced by the Jacobi Identity . Lie algebras abstract the essential nature of Infinitesimal Transformation s, and have become ubiquitous in mathematics. There are other specific types of non-associative structures that have been studied in depth. They tend to come from some specific applications. Some of these arise in for further examples. NON-ASSOCIATIVE ALGEBRAS OVER A FIELD A non-associative algebra (or '''distributive algebra''') over a field ''K'' is a ''K''-vector space ''A'' equipped with a ''K''- Bilinear map . There are left and right multiplication maps and . The ''enveloping algebra'' of ''A'' is the subalgebra of all ''K''-endomorphisms of ''A'' generated by the multiplication maps. An algebra is '' Unital '' or ''unitary'' if it has a Unit or identity element ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra. An algebra is '' Power Associative '' if is well defined for all ''x'' in the algebra and all positive integer ''n'': equivalently the subalgebra generated by any one element is associative. An algebra is '' Alternative '' if (''xx'')''y'' = ''x''(''xy'') and ''y''(''xx'') = (''yx'')''x'' for all ''x'' and ''y'': equivalently the subalgebra generated by any two elements is associative. A Jordan Algebra is Commutative and satisfies the Jordan property (''xy'')(''xx'') = ''x''(''y''(''xx'')) for all ''x'' and ''y''. A Lie Algebra is Anticommutative and satisfies ''xx'' = 0 and the '' Jacobi Identity '' ''x''(''yz'') + ''y''(''zx'') + ''z''(''xy'') = 0. These properties are related by associative implies alternative implies power associative; commutative and associative implies Jordan implies power associative. None of the converse implications hold. REFERENCES
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