Skip to main content
SpringerLink
Log in

Representations and categorical realization of Hom-quasi-Hopf algebras

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

We give a monoidal category approach to Hom-coassociative coalgebra by imposing the Hom-coassociative law up to some isomorphisms on the comultiplication map and requiring that these isomorphisms satisfy the copentagon axiom and obtain a Hom-coassociative 2-coalgebra, which is a 2-category. Second, we characterize Hom-bialgebras in terms of their categories of modules. Finally, we give a categorical realization of Hom-quasi-Hopf algebras using Hom-coassociative 2-coalgebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baez J, Dolan J. Higher-dimensional algebra and topological quantum field theory. J Math Phys, 1995, 36: 6073–6105

    Article  MathSciNet  MATH  Google Scholar 

  2. Caenepeel S, Goyvaerts I. Monoidal Hom-Hopf algebras. Comm Algebra, 2011, 39: 2216–2240

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang W, Wang Z, Wu K, Yang Z. Categorification and quasi-Hopf algebras. Commun Theor Phys, 2011, 56: 207–210

    Article  MATH  Google Scholar 

  4. Cheng Y, Su Y. (Co)Homology and universal central extension of Hom-Leibniz algebras. Acta Math Sin (Engl Ser), 2011, 27(5): 813–830

    MathSciNet  MATH  Google Scholar 

  5. Cheng Y, Yang H. Low-dimensional cohomology of the q-deformed Heisenberg-Virasoro algebra of Hom-type. Front Math China, 2010, 5(4): 607–622

    Article  MathSciNet  MATH  Google Scholar 

  6. Crane L, Frenkel I. Four-dimensional topological quantum field theory, Hopf categories and the canonical bases. J Math Phys, 1994, 35: 5136–5154

    Article  MathSciNet  MATH  Google Scholar 

  7. Drinfeld V. Quasi-Hopf algebras. Leningrad Math J, 1990, 1: 1419–1457

    MathSciNet  Google Scholar 

  8. Elhamdadi M, Makhlouf A. Hom-quasi-bialgebra. arXiv: 1209.0988v1

  9. Etingof P, Nikshych D, Ostrik V. On fusion categories. Ann Math, 2005, 162(2): 581–642

    Article  MathSciNet  MATH  Google Scholar 

  10. Frenkel I, Khovanov M, Stroppel C. A categorification of finite-dimensional irreducible representations of quantum sl 2 and their tensor products. Selecta Math (N S), 2006, 12: 379–431

    Article  MathSciNet  MATH  Google Scholar 

  11. Hartwig J, Larsson D, Silvestrov S. Deformations of Lie algebras using σ-derivations. J Algebra, 2006, 295: 314–361

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang H, Liu G, Ye Y. Quivers, quasi-quantum groups and finite tensor categories. Comm Math Phys, 2011, 303: 595–612

    Article  MathSciNet  MATH  Google Scholar 

  13. Kassel C. Quantum Groups. Grad Texts in Math, Vol 155. New York: Springer-Verlag, 1995

    MATH  Google Scholar 

  14. Larsson D, Silvestrov S. Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities. J Algebra, 2005, 288: 321–344

    Article  MathSciNet  MATH  Google Scholar 

  15. MacLane S. Categories for a Working Mathematician. 2nd ed. Grad Texts in Math, Vol 5. Berlin: Springer, 1998

    Google Scholar 

  16. Majid S. Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras. In: Doebner H, Dobrev V, Ushveridze A, eds. Generalized Symmetries in Physics. Singapore: World Sci, 1994, 13–41

    Google Scholar 

  17. Makhlouf A, Silvestrov S. Hom-algebra and Hom-coalgebras. J Algebra Appl, 2010, 9(4): 553–589

    Article  MathSciNet  MATH  Google Scholar 

  18. Sheng Y. Representation of Hom-Lie algebra. Algebr Represent Theory, 2012, 15(6): 1081–1098

    Article  MathSciNet  MATH  Google Scholar 

  19. Yau D. Hom-algebras and homology. J Lie Theory, 2009, 19: 409–421

    MathSciNet  MATH  Google Scholar 

  20. Yau D. Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. J Phys A: Math Theory, 2009, 42: 165202

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics and Institute of Contemporary Mathematics, Henan University, Kaifeng, 475004, China

    Yongsheng Cheng

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Xiufu Zhang

Authors
  1. Yongsheng Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiufu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongsheng Cheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, Y., Zhang, X. Representations and categorical realization of Hom-quasi-Hopf algebras. Front. Math. China 10, 1263–1281 (2015). https://doi.org/10.1007/s11464-015-0460-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-015-0460-4

Keywords

MSC

Search

Navigation