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][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.
Keywords:
skew-symmetric element, standard involution, Lie algebra, free associative algebra, Jordan derivation, Jordan s-identity.
Citation:
S. R. Sverchkov, “The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras”, Sibirsk. Mat. Zh., 51:3 (2010), 626–637; Siberian Math. J., 51:3 (2010), 496–506
\Bibitem{Sve10}
\by S.~R.~Sverchkov
\paper The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras
\jour Sibirsk. Mat. Zh.
\yr 2010
\vol 51
\issue 3
\pages 626--637
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\jour Siberian Math. J.
\yr 2010
\vol 51
\issue 3
\pages 496--506
\crossref{https://doi.org/10.1007/s11202-010-0052-1}
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Linking options:
https://www.mathnet.ru/eng/smj2113
https://www.mathnet.ru/eng/smj/v51/i3/p626
This publication is cited in the following 2 articles:
Kashuba I. Mathieu O., “On the Free Jordan Algebras”, Adv. Math., 383 (2021), 107690
S. R. Sverchkov, “Jordan s-identities in three variables”, Algebra and Logic, 50:1 (2011), 62–88
Citation in format AMSBIB
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