American Institute of Mathematical Sciences

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

Jordan elements and Left-Center of a Free Leibniz algebra

A. S. Dzhumadil'daev 1,

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.
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.   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

show all references

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

[1]

Randall Dougherty and Thomas Jech. Left-distributive embedding algebras. Electronic Research Announcements, 1997, 3: 28-37.

[2]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
[3]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[4]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
[5]

Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
[6]

Benoît Jubin, Norbert Poncin, Kyosuke Uchino. Free Courant and derived Leibniz pseudoalgebras. Journal of Geometric Mechanics, 2016, 8 (1) : 71-97. doi: 10.3934/jgm.2016.8.71
[7]

Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923
[8]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
[9]

Ünver Çiftçi. Leibniz-Dirac structures and nonconservative systems with constraints. Journal of Geometric Mechanics, 2013, 5 (2) : 167-183. doi: 10.3934/jgm.2013.5.167
[10]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
[11]

A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20.

[12]

Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure & Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95
[13]

Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75.

[14]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems & Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643
[15]

T. J. Newman. Modeling Multicellular Systems Using Subcellular Elements. Mathematical Biosciences & Engineering, 2005, 2 (3) : 613-624. doi: 10.3934/mbe.2005.2.613
[16]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[17]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39
[18]

L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57.

[19]

Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001
[20]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences