Categories of weight modules for unrolled restricted quantum groups at roots of unity

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Resumo

Motivated by connections to the singlet vertex operator algebra in the$\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolledrestricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finitedimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots ofunity with a focus on its category of weight modules. We show that thebraid group action naturally extends to the unrolled quantum groups andthat the category of weight modules is a generically semi-simple ribboncategory (previously known only for odd roots) with trivial Mügercenter and self-dual projective modules.

Sobre autores

Matthew Rupert

Utah State University

Bibliografia

  1. N. Geer, B. Patureau-Mirand, and V. Turaev, “Modified quantum dimensions and re-normalized link invariants”, Compos. Math., 145:1 (2009), 196–212
  2. T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds, and their sets, Ser. Knots Everything, 29, World Sci. Publ., River Edge, NJ, 2002
  3. N. Geer, B. Patureau-Mirand, and V. Turaev, “Modified $6j$-symbols and 3-manifold invariants”, Adv. Math., 228:2 (2011), 1163–1202
  4. N. Geer and B. Patureau-Mirand, “Topological invariants from nonrestricted quantum groups”, Algebr. Geom. Topol., 13:6 (2013), 3305–3363
  5. N. Geer and B. Patureau-Mirand, “The trace on projective representations of quantum groups”, Lett. Math. Phys., 108:1 (2018), 117–140
  6. C. Blanchet, F. Costantino, N. Geer, and B. Patureau-Mirand, “Non semi-simple TQFTs from unrolled quantum $sl(2)$”, Proceedings of the Gökova geometry-topology conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova, 2016, 218–231
  7. M. De Renzi, Non-semisimple extended topological quantum field theories
  8. M. De Renzi, N. Geer, and B. Patureau-Mirand, “Nonsemisimple quantum invariants and TQFTs from small and unrolled quantum groups”, Algebr. Geom. Topol., 20:7 (2020), 3377–3422
  9. D. Adamovic, “Classification of irreducible modules of certain subalgebras of free boson vertex algebra”, J. Algebra, 270:1 (2003), 115–132
  10. D. Adamovic and A. Milas, “On the triplet vertex algebra $W(p)$”, Adv. Math., 217:6 (2008), 2664–2699
  11. D. Adamovic and A. Milas, “Logarithmic intertwining operators and $W(2,2p-1)$ algebras”, J. Math. Phys., 48:7 (2007), 073503
  12. D. Adamovic and A. Milas, “Some applications and constructions of intertwining operators in logarithmic conformal field theory”, Lie algebras, vertex operator algebras, and related topics, Contemp. Math., 695, Amer. Math. Soc., Providence, RI, 2017, 15–27
  13. T. Creutzig and A. Milas, “Higher rank partial and false theta functions and representation theory”, Adv. Math., 314 (2017), 203–227
  14. T. Creutzig, A. Milas, and S. Wood, “On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions”, Int. Math. Res. Not. IMRN, 2017:5 (2017), 1390–1432
  15. T. Creutzig and A. Milas, “False theta functions and the Verlinde formula”, Adv. Math., 262 (2014), 520–545
  16. T. Creutzig, A. Milas, and M. Rupert, “Logarithmic Link invariants of $overline{U}_q^{ H}(mathfrak{sl}_2)$ and asymptotic dimensions of singlet vertex algebras”, J. Pure Appl. Algebra, 222:10 (2018), 3224–3247
  17. T. Creutzig, A. M. Gainutdinov, and I. Runkel, “A quasi-Hopf algebra for the triplet vertex operator algebra”, Commun. Contemp. Math., 22:3 (2020), 1950024
  18. J. Auger, T. Creutzig, S. Kanade, and M. Rupert, “Braided tensor categories related to $mathcal{B}_p$ vertex algebras”, Comm. Math. Phys., 378:1 (2020), 219–260
  19. K. Bringmann and A. Milas, “$W$-algebras, higher rank false theta functions, and quantum dimensions”, Selecta Math. (N.S.), 23:2 (2017), 1249–1278
  20. B. Feigin and I. Tipunin, Logarithmic CFTs connected with simple Lie algebras
  21. A. Milas, “Characters of modules of irrational vertex algebras”, Conformal field theory, automorphic forms and related topics, Contrib. Math. Comput. Sci., 8, Heidelberg, Springer, 2014, 1–29
  22. T. Creutzig, “Logarithmic W-algebras and Argyres–Douglas theories at higher rank”, J. High Energy Phys., 2018:11 (2018), 188
  23. I. Flandoli and S. Lentner, “Logarithmic conformal field theories of type $B_n$, $ell=4$ and symplectic fermions”, J. Math. Phys., 59:7 (2018), 071701
  24. A. M. Gainutdinov, S. Lentner, and T. Ohrmann, Modularization of small quantum groups
  25. S. D. Lentner, “Quantum groups and Nichols algebras acting on conformal field theories”, Adv. Math., 378 (2021), 107517
  26. C. Negron, “Log-modular quantum groups at even roots of unity and the quantum Frobenius I”, Comm. Math. Phys., 382 (2021), 773–814
  27. F. Costantino, N. Geer, and B. Patureau-Mirand, “Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories”, J. Topology, 7:4 (2014), 1005–1053
  28. F. Costantino, N. Geer, and B. Patureau-Mirand, “Some remarks on the unrolled quantum group of $mathfrak{sl}(2)$”, J. Pure Appl. Algebra, 219:8 (2015), 3238–3262
  29. J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category $mathscr{O}$, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008
  30. P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories, Math. Surveys Monogr., 205, Amer. Math. Soc., Providence, RI, 2015
  31. K. Shimizu, “Non-degeneracy conditions for braided finite tensor categories”, Adv. Math., 355 (2019), 106778
  32. S. Lentner, “A Frobenius homomorphism for Lusztig's quantum groups for arbitrary roots of unity”, Commun. Contemp. Math., 18:3 (2016), 1550040
  33. S. Lentner, “The unrolled quantum group inside Lusztig's quantum group of divided powers”, Lett. Math. Phys., 109:7 (2019), 1665–1682
  34. N. Andruskiewitsch and C. Schweigert, “On unrolled Hopf algebras”, J. Knot Theory Ramifications, 27:10 (2018), 1850053
  35. A. Klimyk and K. Schmüdgen, Quantum groups and their representations, Texts Monogr. Phys., Springer-Verlag, Berlin, 1997
  36. V. Chari and A. Pressley, A guide to quantum groups, Corr. reprint of the 1994 original, Cambridge Univ. Press, Cambridge, 1995
  37. M. Jimbo, “A $q$-difference analogue of $U(mathfrak{g})$ and the Yang–Baxter equation”, Lett. Math. Phys., 10:1 (1985), 63–69
  38. G. Lusztig, “Quantum groups at roots of 1”, Geom. Dedicata, 35:1-3 (1990), 89–113
  39. C. De Concini and V. G. Kac, “Representations of quantum groups at roots of 1”, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progr. Math., 92, Birkhäuser Boston, Boston, MA, 1990, 471–506
  40. J. Bichon, “Cosovereign Hopf algebras”, J. Pure Appl. Algebra, 157:2-3 (2001), 121–133
  41. C. Kassel, Quantum groups, Grad. Texts in Math., 155, Springer-Verlag, New York, 1995
  42. N. Geer, B. Patureau-Mirand, and A. Virelizier, “Traces on ideals in pivotal categories”, Quantum Topol., 4:1 (2013), 91–124
  43. P. Etingof, D. Nikshych, and V. Ostrik, “An analogue of Radford's $S^4$ formula for finite tensor categories”, Int. Math. Res. Not. IMRN, 2004:54 (2004), 2915–2933
  44. N. Geer, J. Kujawa, and B. Patureau-Mirand, “Ambidextrous objects and trace functions for nonsemisimple categories”, Proc. Amer. Math. Soc., 141:9 (2013), 2963–2978

Declaração de direitos autorais © Rupert M., 2022

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