Leibniz Algebras Associated with Representations of Euclidean Lie Algebra

  • J. Q. AdashevEmail author
  • B. A. Omirov
  • S. Uguz


In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra \(\mathfrak {e}(2)\) as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by I) as a right \(\mathfrak {e}(2)\)-module is associated to representations of \(\mathfrak {e}(2)\) in \(\mathfrak {sl}_{2}({\mathbb {C}})\oplus \mathfrak {sl}_{2}({\mathbb {C}}), \mathfrak {sl}_{3}({\mathbb {C}})\) and \(\mathfrak {sp}_{4}(\mathbb {C})\). Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra \({\mathfrak {e(n)}}\) as its liezation I being an (n + 1)-dimensional right \({\mathfrak {e(n)}}\)-module defined by transformations of matrix realization of \(\mathfrak {e(n)}\). Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra \(\mathfrak {D}_{k}\) and describe the structure of Leibniz algebras with corresponding Lie algebra \(\mathfrak {D}_{k}\) and with the ideal I considered as a Fock \(\mathfrak {D}_{k}\)-module.


Leibniz algebra Euclidean lie algebra Diamond lie algebra Representation of euclidean lie algebra Fock module 

Mathematics Subject Classification (2010)

17A32 17B10 17B30 


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This work was supported by Agencia Estatal de Investigación (Spain) grant MTM2016-79661-P (European FEDER support included, UE) and by Ministry of Education and Science of the Republic of Kazakhstan the grant No. 0828/GF4.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUzbekistan Academy of SciencesTashkentUzbekistan
  2. 2.National University of UzbekistanTashkentUzbekistan
  3. 3.Department of MathematicsArts and Sciences Faculty Harran UniversityŞanliurfaTurkey

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