Quadratic Jordan algebras and cubing operations
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- by Kevin McCrimmon PDF
- Trans. Amer. Math. Soc. 153 (1971), 265-278 Request permission
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 ${\mathfrak {A}^ + }$.References
- I. N. Herstein and Erwin Kleinfeld, Lie mappings in characteristic $2$, Pacific J. Math. 10 (1960), 843–852. MR 114828, DOI 10.2140/pjm.1960.10.843
- Max Koecher, Eine Charakterisierung der Jordan-Algebren, Math. Ann. 148 (1962), 244–256 (German). MR 144935, DOI 10.1007/BF01470752
- Kevin McCrimmon, A general theory of Jordan rings, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1072–1079. MR 202783, DOI 10.1073/pnas.56.4.1072
- K. McCrimmon and R. D. Schafer, On a class of noncommutative Jordan algebras, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1–4. MR 206063, DOI 10.1073/pnas.56.1.1
- Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470, DOI 10.1007/978-3-642-94947-0
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 153 (1971), 265-278
- MSC: Primary 17.40
- DOI: https://doi.org/10.1090/S0002-9947-1971-0268239-2
- MathSciNet review: 0268239