Skip to main content
SpringerLink
Log in

Representations of Crossed Modules and Other Generalized Yetter-Drinfel’d Modules

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel’d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfel’d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel’d (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators.

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. Alexandre, C., Bordemann, M., Riviere, S., Wagemann, F.: Rack-bialgebras and star-products on duals of Leibniz algebras. ArXiv e-prints (2014)

  2. Bantay, P.: Characters of crossed modules and premodular categories. In: Moonshine: the first quarter century and beyond, London Math. Soc. Lecture Note Ser., vol. 372, pp 1–11. Cambridge University Press, Cambridge (2010)

  3. Beck, J.: Distributive laws. In: Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), pp 119–140. Springer, Berlin (1969)

  4. Brzeziński, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191(2), 467–492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caenepeel, S., Militaru, G., Zhu, S.: Crossed modules and Doi-Hopf modules. Israel J. Math. 100, 221–247 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carter, J.S., Crans, A.S., Elhamdadi, M., Saito, M.: Cohomology of categorical self-distributivity. J. Homotopy Relat. Struct. 3(1), 13–63 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Crans, A., Wagemann, F.: Crossed modules of racks. Homology, Homotopy and its Appl 16(2), 85–106 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crans, A.S.: Lie 2-algebras. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–University of California, Riverside (2004)

  9. Cuvier, C.: Homologie de Leibniz et homologie de Hochschild. C. R. Acad. Sci. Paris Sér. I Math. 313(9), 569–572 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantum groups. In: Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys., vol. 9, pp 97–110. World Sci. Publ., Teaneck, NJ (1989)

  11. Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. In: Algebraic analysis, Vol. I, pp 129–139. Academic Press, Boston, MA (1988)

  12. Fox, T.F., Markl, M.: Distributive laws, bialgebras, and cohomology. In: Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, pp 167–205. Amer. Math. Soc., Providence, RI (1997)

  13. Hennings, M.A.: On solutions to the braid equation identified by Woronowicz. Lett. Math. Phys. 27(1), 13–17 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1), 37–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kraehmer, U., Slevin, P.: Factorisations of distributive laws. ArXiv e-prints (2014)

  16. Lebed, V.: Braided objects: unifying algebraic structures and categorifying virtual braids (2012). Thesis (Ph.D.)–Université Paris 7

  17. Lebed, V.: Braided systems: a unified treatment of algebraic structures with several operations. ArXiv e-prints (2013)

  18. Lebed, V.: Categorical aspects of virtuality and self-distributivity. J. Knot Theory Ramifications 22(9), 1350,045, 32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lebed, V.: Homologies of algebraic structures via braidings and quantum shuffles. J. Algebra 391, 152–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lebed, V.: R-matrices, Yetter-Drinfel’d modules and Yang-Baxter equation. Axioms 2(3), 443–476 (2013)

    Article  MATH  Google Scholar 

  21. Li, F.: The right braids, quasi-braided pre-tensor categories, and general Yang-Baxter operators. Comm. Algebra 32(2), 397–441 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Livernet, M., Mesablishvili, B., Wisbauer, R.: Generalised bialgebras and entwined monads and comonads. J. Pure Appl. Algebra 219(8), 3263–3278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Loday, J.L.: Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301. Springer, Berlin (1992). Appendix E by María O. Ronco

    Google Scholar 

  24. Loday, J.L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. (2) 39(3-4), 269–293 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Loday, J.L.: Generalized bialgebras and triples of operads. Astérisque 320, x+116 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Loday, J.L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296(1), 139–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Maier, J., Schweigert, C.: Modular categories from finite crossed modules. J. Pure Appl. Algebra 215(9), 2196–2208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matveev, S.V.: Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119(161) (1), 78–88, 160 (1982)

    MathSciNet  Google Scholar 

  29. Mesablishvili, B., Wisbauer, R.: Bimonads and Hopf monads on categories. J. K-Theory 7(2), 349–388 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Perk, J.H.H., Au-Yang, H.: Yang-Baxter equations, vol. 5. Elsevier Science, Oxford (2006)

  31. Radford, D.E.: Solutions to the quantum Yang-Baxter equation and the Drinfel’d double. J. Algebra 161(1), 20–32 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Woronowicz, S.L.: Solutions of the braid equation related to a Hopf algebra. Lett. Math. Phys. 23(2), 143–145 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yetter, D.N.: Quantum groups and representations of monoidal categories. Math. Proc. Cambridge Philos. Soc. 108(2), 261–290 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yetter, D.N.: Functorial knot theory, Series on Knots and Everything, vol. 26. World Scientific Publishing Co., Inc., River Edge (2001). Categories of tangles, coherence, categorical deformations, and topological invariants

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de La Houssinière, 44322, Nantes cedex 3, France

    Victoria Lebed & Friedrich Wagemann

Authors
  1. Victoria Lebed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Friedrich Wagemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victoria Lebed.

Additional information

This work was supported by Henri Lebesgue Centre (University of Nantes), and by the program ANR-11-LABX-0020-01.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lebed, V., Wagemann, F. Representations of Crossed Modules and Other Generalized Yetter-Drinfel’d Modules. Appl Categor Struct 25, 455–488 (2017). https://doi.org/10.1007/s10485-015-9421-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-015-9421-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation