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Polynomial Identities

  • Guy Roos
Chapter
Part of the Progress in Mathematics book series (PM, volume 185)

Abstract

Let V be a vector space over \( \mathbb{R} \) (resp. over \( \mathbb{C} \)). We call V a real (resp. Hermitian,resp. complex) Jordan triple system if it is endowed with a triple product
$$ VxVxV \to V $$
$$ (x,y,z) \mapsto \{ xyz\} = \{ x,y,z\} $$
  1. a)

    \( \mathbb{R} \) - trilinear in the real case, \( \mathbb{C} \)-bilinear in (x, z) and \( \mathbb{C} \)-antilinear in y in the Hermitian case, \( \mathbb{C} \)-trilinear in the complex case;

     
  2. b)

    symmetric in \( (x,z):\{ xyz\} = \{ zyx\} \);

     
  3. c)

    satisfying the “Jordan identity:”

     
$$ \{ xy\{ uuz\} \} - \{ uu\{ xyz\} \} = \{ \{ xyu\} uz\} - \{ u\{ uxy\} z\} $$
(J)
.

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Guy Roos
    • 1
  1. 1.Département de MathématiquesUniversité de PoitiersPoitiers CedexFrance

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