Abstract
We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [a, b] and [a, b]3, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zhevlakov K. A., Slinko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, Inc., New York and London (1982) (Pure and Applied Mathematics, 104).
Cohn P., Universal Algebra, Harper and Row, Publishers, New York, Evanston, and London (1965) (Harper’s Series in Modern Mathematics).
McCrimmon K., A Taste of Jordan Algebras, Springer-Verlag, New York (2004).
Smith B. T. and Wos L. T., “An unnatural attack on the structure problem for the free ring on 3 letter: an application on quad arithmetic,” in: Computers in Nonassociative Rings and Algebras, Acad. Press, New York (1977).
Sverchkov S. R., “Quasivariety of special Jordan algebras,” Algebra and Logic, 22, No. 5, 406–414 (1983).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2010 Sverchkov S. R.
To the 70th anniversary of Yu. L. Ershov.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 626–637, May–June, 2010.
Rights and permissions
About this article
Cite this article
Sverchkov, S.R. The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras. Sib Math J 51, 496–506 (2010). https://doi.org/10.1007/s11202-010-0052-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-010-0052-1