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In mathematics, a J-structure is an algebraic structure over a 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 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''(). 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 and the projection onto the first factor, the set of ''g'' which occur, is the ''structure group'' of ''j'', denoted ''G(''j''). A ''J-structure'' is a triple (''V'',''j'',''e'') where ''V'' is a vector space over ''K'', ''j'' is a birational map from ''V'' to itself and ''e'' is a non-zero element of ''V'' satisfying the following conditions.〔Springer (1973) p.10〕 * ''j'' is a homogeneous birational involution of degree −1 * ''j'' is regular at ''e'' and ''j''(''e'') = ''e'' * if ''j'' is regular at ''x'', ''e'' + ''x'' and ''e'' + ''j''(''x'') then : * the orbit ''G'' ''e'' of ''e'' under the structure group ''G'' = ''G''(''j'') is a Zariski open subset of ''V''. The ''norm'' associated to a J-structure (''V'',''j'',''e'') is the numerator ''N'' of ''j'', normalised so that ''N''(''e'') = 1. The ''degree'' of the J-structure is the degree of ''N'' as a homogeneous polynomial map.〔Springer (1973) p.11〕 The ''quadratic map'' of the structure is a map ''P'' from ''V'' to End(''V'') defined in terms of the differential d''j'' at an invertible ''x''.〔Springer (1973) p.16〕 We put : The quadratic map turns out to be a quadratic polynomial map on ''V''. The subgroup of the structure group ''G'' generated by the invertible quadratic maps is the ''inner structure group'' of the J-structure. It is a closed connected normal subgroup.〔Springer (1973) p.18〕 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「J-structure」の詳細全文を読む スポンサード リンク
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