Skip to main content
SpringerLink
Log in

Pivotal Weak Turaev \(\pi\)-Coalgebras

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper, we first introduce the notion of \(\pi\)-pivotal elements in a weak Turaev \(\pi\)-coalgebra \(H\) and show that the representation category of \(H\) is a pivotal crossed category if and only if there is a \(\pi\)-pivotal element in \(H\). Also we discuss the relation between \(\pi\)-pivotal elements and \(\pi\)-ribbon elements of a quasitriangular weak Turaev \(\pi\)-coalgebra. Finally, we obtain a generalized Deligne Type theorem.

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. V. G. Turaev, “Crossed group-categories,” Arab. J. Sci. Eng. Sec. C Theme Issues 33 (2C), 483–503 (2008).

    MathSciNet  MATH  Google Scholar 

  2. P. Freyd and D. Yetter, “Braided compact closed categories with applications to low-dimensional topology,” Adv. Math. 77 (2), 156–182 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Virelizier, “Hopf group-coalgebras,” J. Pure Appl. Algebra 171 (1), 75–122 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Zunino, “Yetter–Drinfeld modules for crossed structures,” J. Pure Appl. Algebra 193 (1-3), 313–343 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Zunino, “Double construction for crossed Hopf coalgebras,” J. Algebra 278 (1), 43–75 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. H. Wang, “Coquasitriangular Hopf group algebras and Drinfel’d co-doubles,” Comm. Algebra 35 (1), 77–101 (2009).

    Article  MATH  Google Scholar 

  7. S. H. Wang, “New Turaev braided group categories and group Schur–Weyl duality,” Appl. Categ. Struct. 21 (2), 141–166 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. H. Wang, “Turaev group coalgebras and twisted Drinfeld double,” Indiana Univ. Math. J. 58 (3), 1395–1417 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. Q. G. Chen and D. G. Wang, “A class of coquasitriangular Hopf group algebras,” Comm. Algebra 44 (1), 310–335 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  10. Q. G. Chen and D. G. Wang, “Constructing New crossed group categories over weak Hopf group algebras,” Math. Slovaca 65 (3), 473–492 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. Q. G. Chen and D. G. Wang, “Partial group (co)actions of Hopf group coalgebras,” Filomat 28 (1), 131–151 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  12. X. L. Fang and S. H. Wang, “New Turaev braided group categories and group corings based on quasi-Hopf group coalgebras,” Comm. Algebra 41 (11), 4195–4226 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  13. D. G. Wang and Q. G. Chen, “A Maschke type theorem for weak group entwined modules and applications,” Israel J. Math. 204 (1), 329–358 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. D. G. Wang and D. W. Lu, “Ore extensions of Hopf group coalgebras,” J. Korean Math. Soc. 51 (2), 325–344 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. G. Turaev and A. Virelizier, “On the graded center of graded categories,” J. Pure Appl. Algebra 217 (10), 1895–1941 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Van Daele and S. H. Wang, “New braided crossed categories and Drinfel’d quantum double for weak Hopf group coalgebras,” Comm. Algebra 36 (6), 2341–2386 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Virelizier, “Involutory Hopf group-coalgebras and flat bundles over 3-manifolds,” Fund. Math. 188 (1), 241–270 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Yang and S. H. Wang, “Constructing new braided T-categories over regular multiplier Hopf algebra,” Comm. Algebra 39 (9), 3073–3089 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Selinger, “A survey of graphical languages for monoidal categories,” in New Structures for Physics, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2010), Vol. 813.

    Chapter  Google Scholar 

  20. D. N. Yetter, “Framed tangles and a theorem of Deligne on braided deformations of tannakian categories,” in Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics (American Mathematical Society, New York, 1992), Vol. 134, pp. 325–345.

    Chapter  Google Scholar 

  21. J. Bichon, “Cosovereign Hopf algebras,” J. Pure Appl. Algebra 157 (2–3), 121–133 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Bruguières and A. Virelizier, “Hopf monads,” Adv. Math. 215 (2), 679–733 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Panaite, “Ribbon and charmed elements for quasitriangular Hopf algebras,” Comm. Algebra 25 (3), 973–977 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  24. X. H. Zhang, X. F. Zhao, and S. H. Wang, “Sovereign and ribbon weak Hopf algebras,” Kodai Math. J. 38 (2), 451–469 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. L. H. Kauffman and D. E. Radford, “A necessary and sufficient condition for a finite-dimensional Drinfel’d double to be a ribbon Hopf algebra,” J. Algebra 159 (1), 98–114 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Drabant, “Notes on balanced categories and Hopf algebras,” in Quantum Groups and Lie Theory, London Mathematical Society Lecture Note Series (London Mathematical Society, London, 2002), Vol. 290, pp. 63–88.

    MATH  Google Scholar 

  27. P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (New York, American Mathematical Society, 2015).

    Book  MATH  Google Scholar 

  28. V. G. Turaev, Homotopy Quantum Field Theory, EMS Tracts in Math. 10 (European Mathematical Society, Zürich, 2010).

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their deep gratitude to the anonymous referee for a careful reading and valuable suggestions.

Funding

The work of X. H. Zhang was supported by the National Natural Science Foundation of China (Nos. 11871301 and 12271292) and the Taishan Scholar Project of Shandong Province (No. tsqn202103060). The work of S. J. Guo was supported by the NSF of China (No. 12161013). The work of S. X. Wang was supported by the Anhui Provincial Natural Science Foundation (No. 1908085MA03), the Key University Science Research Project of Anhui Province (No. KJ2020A0711), and the NSF of Chuzhou University (No. 2021qd08).

Author information

Authors and Affiliations

  1. Qufu Normal University, Qufu, 273165, China

    Xiaohui Zhang

  2. Guizhou University of Finance and Economics, Guiyang, 550025, China

    Shuangjian Guo

  3. Chuzhou University, Chuzhou, 239000, China

    Shengxiang Wang

Authors
  1. Xiaohui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuangjian Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shengxiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaohui Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Guo, S. & Wang, S. Pivotal Weak Turaev \(\pi\)-Coalgebras. Math Notes 112, 797–815 (2022). https://doi.org/10.1134/S0001434622110153

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434622110153

Keywords

Search

Navigation