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A pair of Jordan triple systems of Jordan pair type. (English) Zbl 1084.17508
Summary: A Jordan pair \(V=(V^+,V^-)\) is a pair of modules over a unitary commutative associative ring \(K\), together with a pair \((Q_+,Q_-)\) of quadratic mappings \(Q_\sigma:V^\sigma\to\operatorname{Hom}_K(V^{-\sigma},V^\sigma),\sigma=\pm\), so that the following identities and their liniarizations are fulfilled for \(\sigma=\pm\):
JP1 \(D_\sigma(x,y)Q_\sigma(x)=Q_\sigma(x)D_{-\sigma}(y,x)\),
JP2 \(D_\sigma(Q_\sigma(x)y,y)=D_\sigma(x,Q_{-\sigma}(y)x)\),
JP3 \(Q_\sigma(Q_\sigma(x)y)=Q_\sigma(x)Q_{-\sigma}(y)Q_\sigma(x).\)
Here, \(D_\sigma(x,y)z=Q_\sigma(x,z)y:=Q_\sigma(x+z)y-Q_\sigma(x)y-Q_\sigma(z)y.\)
According to an example with quadratic mappings \(Q_\sigma:V^\sigma\to\text{End}(V^{-\sigma})\), we get a little different approach and we define the concept of pair of Jodan Triple Systems of Jordan Pair type.
MSC:
17C99 Jordan algebras (algebras, triples and pairs)
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