Please wait a minute...
Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Front. Math. China    2017, Vol. 12 Issue (1) : 19-33    https://doi.org/10.1007/s11464-016-0590-3
RESEARCH ARTICLE
Weak rigid monoidal category
Haijun CAO()
School of Science, Shandong Jiaotong University, Jinan 250375, China
 Download: PDF(178 KB)  
 Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

We define the right regular dual of an object X in a monoidal category C , and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End(F) whenever (F, J) is a fiber functor from category C to V ec and every XC has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.

Keywords Semilattice graded weak Hopf algebra      regular right dual      weak rigid monoidal category     
Corresponding Author(s): Haijun CAO   
Issue Date: 17 November 2016
 Cite this article:   
Haijun CAO. Weak rigid monoidal category[J]. Front. Math. China, 2017, 12(1): 19-33.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0590-3
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I1/19
1 Cartier P. A primer of Hopf algebras. In: Cartier P, Julia B, Moussa P, Vanhove P, eds. Frontiers in Number Theory, Physics and Geometry II. Berlin: Springer, 2007, 537–615
https://doi.org/10.1007/978-3-540-30308-4_12
2 Deligne P. Catégories tannakiennes. In: The Grothendieck Festschrift, Vol II. Progr Math, Vol 87. Basel: Birkhäuser, 1990, 111–196
3 Deligne P, Milne J. Tannakian categories. In: Lecture Notes in Math, Vol 900. Berlin: Springer, 1982, 101–228
4 Drinfield V. Quasi-Hopf algebras. Leningrad Math J, 1989, 1: 1419–1457
5 Etingof P, Schiffmann O. Lectures on Quantum Groups.Somerville: International Press, 1980
6 Hopf H. Über die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen. Ann of Math, 1941, 42: 22–52
https://doi.org/10.2307/1968985
7 Kelly G. On Mac Lane’s condition for coherence of natural associativities, commutativities, etc. J Algebra, 1964, 1: 397–402
https://doi.org/10.1016/0021-8693(64)90018-3
8 Krein M. A principle of duality for a bicompact group and square block algebra. Dokl Akad Nauk SSSR, 1949, 69: 725–728
9 Li F. Weak Hopf algebras and some new solutions of quantum Yang-Baxter equation. J Algebra, 1998, 208: 72–100
https://doi.org/10.1006/jabr.1998.7491
10 Li F, Cao H J. Semilattice graded weak Hopf algebra and its related quantum G-double. J Math Phys, 2005, 46(8): 1–17
https://doi.org/10.1063/1.2000687
11 Mac Lane S. Natural associativity and commutativity. Rice Univ Studies, 1963, 49(4): 28–46
12 Mac Lane S. Categories for the working mathematician. 2nd ed. Grad Texts in Math, Vol 5. Berlin: Springer, 1998
13 Majid S. Foundations of Quantum Group Theory.Cambridge: Cambridge Univ Press, 2000
14 Saavedra Rivano N. Catégories Tannakiennes. Lecture Notes in Math, Vol 265. Berlin: Springer, 1972
15 Tannaka T. Über den Dualitätssatz der nichtkommutativen topologischen Gruppen. Tohoku Math J, 1939, 45: 1–12
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed