Jordan elements and Left-Center of a Free Leibniz algebra

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January 2011, 18: 31-49. doi: 10.3934/era.2011.18.31

Jordan elements and Left-Center of a Free Leibniz algebra

1.	Kazakh-British University, Almaty, Kazakhstan

Received February 2011 Revised April 2011 Published July 2011

Abstract
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An element of a free Leibniz algebra is called Jordan if it belongs to a free Leibniz-Jordan subalgebra. Elements of the Jordan commutant of a free Leibniz algebra are called weak Jordan. We prove that an element of a free Leibniz algebra over a field of characteristic 0 is weak Jordan if and only if it is left-central. We show that free Leibniz algebra is an extension of a free Lie algebra by left-center. We find the dimensions of the homogeneous components of the Jordan commutant and the base of its multilinear part. We find criterion for an element of free Leibniz algebra to be Jordan.

Keywords: Leibniz algebras, Jordan elements, left-center..

Mathematics Subject Classification: Primary 17A32; Secondary 17D2.

Citation: A. S. Dzhumadil'daev. Jordan elements and Left-Center of a Free Leibniz algebra. Electronic Research Announcements, 2011, 18: 31-49. doi: 10.3934/era.2011.18.31

References:

[1]	A. S. Dzhumadil'daev, $q$-Leibniz algebras,, Serdica Math., 34 (2008), 415.
[2]	N. Jacobson, "Structure and Representations of Jordan Algebras,", AMS Colloq. Publ., 39 (1968).
[3]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Liebniz algebras and (co)homology,, Math. Ann., 296 (1993), 139. doi: 10.1007/BF01445099.
[4]	J.-L. Loday, Cup-product for Leibniz cohomology and dual Leibniz algebras,, Math. Scand., 77 (1995), 189.
[5]	C. Reutenauer, "Free Lie Algebras,", London Mathematical Society Monographs, (1993).

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References:

[1]	A. S. Dzhumadil'daev, $q$-Leibniz algebras,, Serdica Math., 34 (2008), 415.
[2]	N. Jacobson, "Structure and Representations of Jordan Algebras,", AMS Colloq. Publ., 39 (1968).
[3]	J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Liebniz algebras and (co)homology,, Math. Ann., 296 (1993), 139. doi: 10.1007/BF01445099.
[4]	J.-L. Loday, Cup-product for Leibniz cohomology and dual Leibniz algebras,, Math. Scand., 77 (1995), 189.
[5]	C. Reutenauer, "Free Lie Algebras,", London Mathematical Society Monographs, (1993).

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