Hom-Lie Admissible Hom-Coalgebras and Hom-Hopf Algebras

The aim of this paper is to develop the coalgebra counterpart of the notions introduced by the authors in a previous paper, we introduce the notions of Hom-coalgebra, Hom-coassociative coalgebra and G-Hom-coalgebra for any subgroup G of permutation group S script>3. Also we extend the concept of Lie-admissible coalgebra by Goze and Remm to Hom-coalgebras and show that G-Hom-coalgebras are Hom-Lie admissible Hom-coalgebras, and also establish duality correspondence between classes of G-Hom-coalgebras and G-Hom-algebras. In another hand, we provide relevant definitions and basic properties of Hom-Hopf algebras generalizing the classical Hopf algebras and define the module and comodule structure over Hom-associative algebra or Hom-coassociative coalgebra.


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  1. 1.
    Drinfel'd V. G.: Hopf algebras and the quantum Yang—Baxter equation, Soviet Math. Doklady 32, 254–258 (1985)Google Scholar
  2. 2.
    Goze M., Remm E.: Lie-admissible coalgebras, J. Gen. Lie Theory Appl. 1, no. 1, 19–28 (2007)MATHMathSciNetGoogle Scholar
  3. 3.
    Guichardet A.: Groupes quantiques, InterEditions / CNRS Editions, Paris (1995)MATHGoogle Scholar
  4. 4.
    Hartwig J. T., Larsson D., Silvestrov S. D.: Deformations of Lie algebras using σ-derivations, J. Algebra 295, 314–361 (2006)MATHMathSciNetGoogle Scholar
  5. 5.
    Hellström L., Silvestrov S. D.: Commuting elements in q-deformed Heisenberg algebras, World Scientific, Singapore (2000)MATHGoogle Scholar
  6. 6.
    Kassel C.: Quantum groups, Graduate Text in Mathematics, Springer, Berlin (1995)Google Scholar
  7. 7.
    Larsson D., Silvestrov S. D.: Quasi-hom-Lie algebras, Central Extensions and 2-cocycle-like identities, J. Algebra 288, 321–344 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Larsson D., Silvestrov S. D.: Quasi-Lie algebras, in “Noncommutative Geometry and Representation Theory in Mathematical Physics”, Contemp. Math. 391, Amer. Math. Soc., Providence, RI, 241–248 (2005)MathSciNetGoogle Scholar
  9. 9.
    Larsson D., Silvestrov S. D.: Quasi-deformations of sl script>2(F) using twisted derivations, Comm. Algebra 35, 4303–4318 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Majid S.: Foundations of quantum group theory, Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  11. 11.
    Makhlouf A.: Degeneration, rigidity and irreducible components of Hopf algebras, Algebra Colloquium 12(2), 241–254 (2005)MATHMathSciNetGoogle Scholar
  12. 12.
    Makhlouf A., Silvestrov S. D.: Hom-algebra structures, J. Gen. Lie Theory, Appl. 2(2), 51–64 (2008)MathSciNetGoogle Scholar
  13. 13.
    Montgomery S.: Hopf algebras and their actions on rings, AMS Regional Conference Series in Mathematics 82, (1993)Google Scholar
  14. 14.
    Yau D.: Enveloping algebra of Hom-Lie algebras, J. Gen. Lie Theory Appl. 2(2), 95–108 (2008)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques, Informatique et ApplicationsUniversité de Haute AlsaceMulhouseFrance

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