RESEARCH ARTICLE

Representations and categorical realization of Hom-quasi-Hopf algebras

  • Yongsheng CHENG , 1 ,
  • Xiufu ZHANG 2
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  • 1. School of Mathematics and Statistics and Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, China
  • 2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received date: 05 Jul 2014

Accepted date: 05 Feb 2015

Published date: 12 Oct 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Yongsheng CHENG , Xiufu ZHANG . Representations and categorical realization of Hom-quasi-Hopf algebras[J]. Frontiers of Mathematics in China, 2015 , 10(6) : 1263 -1281 . DOI: 10.1007/s11464-015-0460-4

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

DOI

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

DOI

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

DOI

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

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

DOI

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

DOI

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

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

DOI

10
Frenkel I, Khovanov M, Stroppel C. A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products. Selecta Math (N S), 2006, 12: 379−431

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

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

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

DOI

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

DOI

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

20
Yau D. Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular b<?Pub Caret?>ialgebras. J Phys A: Math Theory, 2009, 42: 165202

DOI

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