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J-structure
In mathematics, a J-structure is an algebraic structure over a field (algebra), field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic of a field, characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras. Definition Let ''V'' be a finite-dimensional vector space over a field ''K'' and ''j'' a rational map from ''V'' to itself, expressible in the form ''n''/''N'' with ''n'' a polynomial map from ''V'' to itself and ''N'' a polynomial in ''K''[''V'']. Let ''H'' be the subset of GL(''V'') × GL(''V'') containing the pairs (''g'',''h'') such that ''g''∘''j'' = ''j''∘''h'': it is a closed subgroup of the product an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan algebra is also denoted ''x'' ∘ ''y'', particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative, meaning that x^n = x \cdots x is independent of how we parenthesize this expression. They also imply that x^m (x^n y) = x^n(x^m y) for all positive integers ''m'' and ''n''. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element x, the operations of multiplying by powers x^n all commute. Jordan algebras were introduced by in an effort to formalize the notion of an algebra of observables in quantum electrodynamics. It was soon shown that the algebras were not useful in this context, howev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |