Skip to main content
SpringerLink
Log in

Remarks on the derived center of small quantum groups

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

Let \({\mathsf {u}}_q(\mathfrak {g})\) be the small quantum group associated with a complex semisimple Lie algebra \(\mathfrak {g}\) and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of \(\mathfrak {g}\) on the center of \({\mathsf {u}}_q(\mathfrak {g})\) and on the higher derived center of \({\mathsf {u}}_q(\mathfrak {g})\). Based on the triviality of this action for \(\mathfrak {g}= \mathfrak {sl}_2, \mathfrak {sl}_3, \mathfrak {sl}_4\), we conjecture that, in finite type A, central elements of the small quantum group \({\mathsf {u}}_q(\mathfrak {sl}_n)\) arise as the restriction of central elements in the big quantum group \({\mathsf {U}}_q(\mathfrak {sl}_n)\). We also study the role of an ideal \({\mathsf {z}}_\mathrm{Hig}\) known as the Higman ideal in the center of \({\mathsf {u}}_q(\mathfrak {g})\). We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of \({\mathsf {z}}_\mathrm{Hig}\) in type A. As an illustration we provide a detailed explicit description of the derived center of \({\mathsf {u}}_q(\mathfrak {sl}_2)\) and its various symmetries.

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

Fig. 1

Similar content being viewed by others

Notes

  1. The extended modular group action restricts to a projective action of the modular group on the Hochschild cohomology of \({\mathsf {u}}\), as was noticed in [24, 35].

  2. This result is first stated in [24], and the authors would like to thank C. Schweigert for pointing out the references to us.

References

  1. Arkhipov, S., Bezrukavnikov, R., Ginzburg, V.: Quantum groups, the loop Grassmannian, and the Springer resolution. J. Am. Math. Soc. 17(3), 595–678 (2004). arXiv:math/0304173

  2. Andersen, H.H.: Tensor products of quantized tilting modules. Commun. Math. Phys. 149(1), 149–159 (1992)

    Article  MathSciNet  Google Scholar 

  3. Andersen, H.H., Polo, P., Wen, K.X.: Representations of quantum algebras. Invent. Math. 104(1), 1–59 (1991)

    Article  MathSciNet  Google Scholar 

  4. Andersen, H.H., Polo, P., Wen, K.X.: Injective modules for quantum algebras. Am. J. Math. 114(3), 571–604 (1992)

    Article  MathSciNet  Google Scholar 

  5. Brown, K.A., Gordon, I.: The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras. Math. Z. 238(4), 733–779 (2001). arXiv:math/9911234

  6. Backelin, E., Kremnizer, K.: Localization for quantum groups at a root of unity. J. Am. Math. Soc. 21(4), 1001–1018 (2008). arXiv:math/0407048

  7. Backelin, E., Kremnizer, K.: Singular localization of \(\mathfrak{g}\)-modules and applications to representation theory. J. Eur. Math. Soc. 17(11), 2763–2787 (2015). arXiv:1011.0896

  8. Bezrukavnikov, R., Lachowska,A.: The small quantum group and the Springer resolution. In: Quantum groups, volume 433 of Contemp. Math., pp. 89–101. American Mathematical Society, Providence, RI (2007). arXiv:math/0609819

  9. Calaque, D., Van den Bergh, M.: Hochschild cohomology and Atiyah classes. Adv. Math. 224(5), 1839–1889 (2010). arXiv:0708.2725

  10. Cohen, M., Westreich, S.: Characters and a Verlinde-type formula for symmetric Hopf algebras. J. Algebra 320(12), 4300–4316 (2008)

    Article  MathSciNet  Google Scholar 

  11. Drinfeld, V.G.: Almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989)

    MathSciNet  Google Scholar 

  12. Feigin, B.L., Gainutdinov, A.M., Semikhatov, A.M., Tipunin, I.Y.: Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center. Comm. Math. Phys. 265(1), 47–93 (2006). arXiv:hep-th/0504093

  13. Gelfand, S., Kazhdan, D.: Examples of tensor categories. Invent. Math. 109(3), 595–617 (1992)

    Article  MathSciNet  Google Scholar 

  14. Ginzburg, V., Kumar, S.: Cohomology of quantum groups at roots of unity. Duke Math. J. 69(1), 179–198 (1993)

    Article  MathSciNet  Google Scholar 

  15. Gorsky, E., Mazin, M., Vazirani, M.: Affine permutations and rational slope parking functions. In: 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), Discrete Math. Theor. Comput. Sci. Proc., AT, pages 887–898. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2014). arXiv:1403.0303

  16. Haboush, W.J.: Central differential operators on split semisimple groups over fields of positive characteristic. In: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, 32ème année (Paris, 1979), volume 795 of Lecture Notes in Math., pp. 35–85. Springer, Berlin (1980)

  17. Haiman, M.D.: Conjectures on the quotient ring by diagonal invariants. J. Algebraic Combin. 3(1), 17–76 (1994)

    Article  MathSciNet  Google Scholar 

  18. Kerler, T.: Mapping class group actions on quantum doubles. Commun. Math. Phys. 168(2), 353–388 (1995). arXiv:hep-th/9402017

  19. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. II. J. Am. Math. Soc. 6(4), 905–947 (1993)

    Article  MathSciNet  Google Scholar 

  20. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. II. J. Am. Math. Soc. 6(4), 949–1011 (1993)

    Article  MathSciNet  Google Scholar 

  21. Lachowska, A.: A counterpart of the Verlinde algebra for the small quantum group. Duke Math. J. 118(1), 37–60 (2003). arXiv:math/0107136

  22. Lachowska, A.: On the center of the small quantum group. J. Algebra 262(2), 313–331 (2003). arXiv:math/0107098

  23. Lyubashenko, V., Majid, S.: Braided groups and quantum Fourier transform. J. Algebra 166(3), 506–528 (1994)

    Article  MathSciNet  Google Scholar 

  24. Lentner, S., Mierach, S..N., Schweigert, C., Sommerhäuser, Y.: Hochschild cohomology and the modular group. J. Algebra 507, 400–420 (2018). arXiv:1707.04032

  25. Lachowska, A., Qi, Y.: The center of the small quantum group I: the principal block in type A. Int. Math. Res. Not. IMRN, (20), 6349–6405 (2018). arXiv:1604.07380

  26. Lachowska, A., Qi, Y.: The center of the small quantum group II: singular blocks. Proc. Lond. Math. Soc. (3) 118, 513–544 (2018). arXiv:1703.02457

  27. Lusztig, G.: Modular representations and quantum groups. In: Classical groups and related topics (Beijing, 1987), volume 82 of Contemp. Math., pp. 59–77. American Mathematical Society, Providence, RI (1989)

  28. Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra. J. Am. Math. Soc. 3(1), 257–296 (1990)

    MathSciNet  MATH  Google Scholar 

  29. Lusztig, G.: Quantum groups at roots of \(1\). Geom. Dedicata. 35(1–3), 89–113 (1990)

    MathSciNet  MATH  Google Scholar 

  30. Lyubashenko, V.: Invariants of \(3\)-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Commun. Math. Phys. 172(3), 467–516 (1995). arXiv:hep-th/9405167

  31. Ostrik, V.: Decomposition of the adjoint representation of the small quantum \({\rm sl}_2\). Commun. Math. Phys. 186(2), 253–264 (1997). arXiv:q-alg/9512026

  32. Qi, Y.: Hopfological algebra. Compos. Math. 150(01), 1–45 (2014). arXiv:1205.1814

  33. Qi, Y.: Morphism spaces in stable categories of Frobenius algebras. Commun. Algebra 47(8), 3239–3249 (2019). arXiv:1801.07838

  34. Radford, D.E.: The trace function and Hopf algebras. J. Algebra 163(3), 583–622 (1994)

    Article  MathSciNet  Google Scholar 

  35. Schweigert, C., Woike, L.: The Hochschild complex of a finite tensor category. (2019). arXiv:1910.00559

  36. Sweedler, M.E.: Integrals for Hopf algebras. Ann. Math. 2(89), 323–335 (1969)

    Article  MathSciNet  Google Scholar 

  37. Zimmermann, A.: Representation theory, volume 19 of Algebra and Applications. Springer, Cham. A homological algebra point of view (2014)

Download references

Acknowledgements

A. L. would like to thank Azat Gainutdinov for useful discussions about the \(\mathfrak {g}\)-action on the center of a small quantum group and on the derived center of the small quantum \(\mathfrak {sl}_2\). Part of this work was carried out during the first author’s visit to California Institute of Technology (Caltech). Both authors thank Caltech for its hospitality and support. A. L. is grateful for the support from Facutlé des Sciences de Base at École Polytechnique Fédérale de Lausanne. Y. Q. is partially supported by the National Science Foundation DMS-1947532 while working on the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, École Polytechnique Fédérale de Lausanne, SB MATH TAN, MA C3 535 (Bâtiment MA), Station 8, 1015, Lausanne, Switzerland

    Anna Lachowska

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    You Qi

Authors
  1. Anna Lachowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. You Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to You Qi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lachowska, A., Qi, Y. Remarks on the derived center of small quantum groups. Sel. Math. New Ser. 27, 68 (2021). https://doi.org/10.1007/s00029-021-00686-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00029-021-00686-7

Mathematics Subject Classification

Search

Navigation