# American Institute of Mathematical Sciences

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

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.
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.   Google Scholar [2] N. Jacobson, "Structure and Representations of Jordan Algebras,", AMS Colloq. Publ., 39 (1968).   Google Scholar [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.  Google Scholar [4] J.-L. Loday, Cup-product for Leibniz cohomology and dual Leibniz algebras,, Math. Scand., 77 (1995), 189.   Google Scholar [5] C. Reutenauer, "Free Lie Algebras,", London Mathematical Society Monographs, (1993).   Google Scholar

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##### References:
 [1] A. S. Dzhumadil'daev, $q$-Leibniz algebras,, Serdica Math., 34 (2008), 415.   Google Scholar [2] N. Jacobson, "Structure and Representations of Jordan Algebras,", AMS Colloq. Publ., 39 (1968).   Google Scholar [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.  Google Scholar [4] J.-L. Loday, Cup-product for Leibniz cohomology and dual Leibniz algebras,, Math. Scand., 77 (1995), 189.   Google Scholar [5] C. Reutenauer, "Free Lie Algebras,", London Mathematical Society Monographs, (1993).   Google Scholar

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