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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic Jordan algebras and cubing operations

Author: Kevin McCrimmon
Journal: Trans. Amer. Math. Soc. 153 (1971), 265-278
MSC: Primary 17.40
MathSciNet review: 0268239
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Abstract: In this paper we show how the Jordan structure can be derived from the squaring and cubing operations in a quadratic Jordan algebra, and give an alternate axiomatization of unital quadratic Jordan algebras in terms of operator identities involving only a single variable. Using this we define nonunital quadratic Jordan algebras and show they can be imbedded in unital algebras. We show that a noncommutative Jordan algebra $ \mathfrak{A}$ (over an arbitrary ring of scalars) determines a quadratic Jordan algebra $ {\mathfrak{A}^ + }$.

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Keywords: Jordan algebra, noncommutative Jordan algebra, quadratic Jordan algebra
Article copyright: © Copyright 1971 American Mathematical Society