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In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by . The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic. If 2 is invertible in the field of coefficients, the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras. ==Definition== A quadratic Jordan algebra consists of a vector space ''A'' over a field ''K'' with a distinguished element 1 and a quadratic map of ''A'' into the ''K''-endomorphisms of ''A'', ''a'' ↦ ''Q''(''a''), satisfying the conditions: * ''Q''(1) = id; * ''Q''(''Q''(''a'')''b'') = ''Q''(''a'')''Q''(''b'')''Q''(''a'') ("fundamental identity"); * ''Q''(''a'')''R''(''b'',''a'') = ''R''(''a'',''b'')''Q''(''a'') ("commutation identity"), where ''R''(''a'',''b'')''c'' = (''Q''(''a'' + ''c'') − ''Q''(''a'') − ''Q''(''c''))''b''. Further, these properties are required to hold under any extension of scalars.〔Racine (1973) p.1〕 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「Quadratic Jordan algebra」の詳細全文を読む スポンサード リンク
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