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  • V - Jordan algebras

  • Skip Garibaldi, Institute for Defense Analyses, USA, Holger P. Petersson, FernUniversität in Hagen, Michel L. Racine, University of Ottawa
  • Book:
    Albert Algebras over Commutative Rings
    Published online:
    07 November 2024
    Print publication:
    21 November 2024, pp 252-316

  • Summary

    This chapter will develop from scratch the elementary theory of (quadratic) Jordan algebras over commutative rings. After a brief account of linear Jordan algebras and their most rudimentary properties over rings in which 2 is invertible, we proceed to para-quadratic algebras, which play the same role in the quadratic setting as is played by ordinary nonassociative algebras in the linear setting. Quadratic Jordan algebras are introduced. We derive a wide range of useful identities and acquaint the reader with the standard examples of special Jordan algebras, namely the Jordan algebra constructed from a unital associative algebra, from an associative algebra with involution, or from a pointed quadratic module. After a brief interlude concerning a peculiar class of two-variable identities, we investigate what are arguably the most important concepts of the theory: invertibility, isotopy, and the structure group. The chapter concludes with a concise description of the Peirce decomposition relative to an idempotent, and also relative to a complete orthogonal system of idempotents.