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Weak rigid monoidal category

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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 X ∈ C has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.

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References

  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

    Google Scholar 

  2. Deligne P. Catégories tannakiennes. In: The Grothendieck Festschrift, Vol II. Progr Math, Vol 87. Basel: Birkhäuser, 1990, 111–196

    MATH  Google Scholar 

  3. Deligne P, Milne J. Tannakian categories. In: Lecture Notes in Math, Vol 900. Berlin: Springer, 1982, 101–228

    Google Scholar 

  4. Drinfield V. Quasi-Hopf algebras. Leningrad Math J, 1989, 1: 1419–1457

    MathSciNet  Google Scholar 

  5. Etingof P, Schiffmann O. Lectures on Quantum Groups. Somerville: International Press, 1980

    MATH  Google Scholar 

  6. Hopf H. Über die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeiner-ungen. Ann of Math, 1941, 42: 22–52

    Article  MathSciNet  MATH  Google Scholar 

  7. Kelly G. On Mac Lane's condition for coherence of natural associativities, commuta-tivities, etc. J Algebra, 1964, 1: 397–402

    Article  MathSciNet  Google Scholar 

  8. Krein M. A principle of duality for a bicompact group and square block algebra. Dokl Akad Nauk SSSR, 1949, 69: 725–728

    MathSciNet  Google Scholar 

  9. Li F. Weak Hopf algebras and some new solutions of quantum Yang-Baxter equation. J Algebra, 1998, 208: 72–100

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  11. Mac Lane S. Natural associativity and commutativity. Rice Univ Studies, 1963, 49(4): 28–46

    MathSciNet  MATH  Google Scholar 

  12. Mac Lane S. Categories for the working mathematician. 2nd ed. Grad Texts in Math, Vol 5. Berlin: Springer, 1998

    MATH  Google Scholar 

  13. Majid S. Foundations of Quantum Group Theory. Cambridge: Cambridge Univ Press, 2000

    MATH  Google Scholar 

  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

    MATH  Google Scholar 

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  1. School of Science, Shandong Jiaotong University, Jinan, 250375, China

    Haijun Cao

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  1. Haijun Cao
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Cao, H. Weak rigid monoidal category. Front. Math. China 12, 19–33 (2017). https://doi.org/10.1007/s11464-016-0590-3

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  • DOI: https://doi.org/10.1007/s11464-016-0590-3

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