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M-traces in (Non-Unimodular) Pivotal Categories

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Abstract

We generalize the notion of a modified trace (or m-trace) to the setting of non-unimodular categories. M-traces are known to play an important role in low-dimensional topology and representation theory, as well as in studying the category itself. Under mild conditions we give existence and uniqueness results for m-traces in pivotal categories.

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References

  1. Alperin, J.L.: Local representation theory, volume 11 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511623592. Modular representations as an introduction to the local representation theory of finite groups (1986)

  2. Andruskiewitsch, N., Schweigert, C.: On unrolled Hopf algebras. arXiv:1701.00153 (2017)

  3. Beliakova, A., Blanchet, C., Gainutdinov, A.M.: Modified trace is a symmetrised integral. arXiv:1801.00321 (2017)

  4. Beliakova, A., Blanchet, C., Geer, N.: Logarithmic Hennings invariants for restricted quantum sl(2). arXiv:1705.03083 (2017)

  5. Blanchet, C., Costantino, F., Geer, N., Patureau-Mirand, B.: Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants. Adv. Math. 301:1–78. https://doi.org/10.1016/j.aim.2016.06.003 (2016)

  6. Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6):1183–1205, 1337. https://doi.org/10.1070/IM1990v035n03ABEH000716 (1989)

  7. Boe, B.D., Kujawa, J.R., Nakano., D.K.: Complexity for modules over the classical Lie superalgebra \(\mathfrak {gl}(m|n)\). Compos. Math. 148(5):1561–1592. https://doi.org/10.1112/S0010437X12000231 (2012)

  8. Barrett, J.W., Westbury, B.W.: Spherical categories. Adv. Math. 143(2):357–375. https://doi.org/10.1006/aima.1998.1800 (1999)

  9. Creutzig, T., Gannon, T.: Logarithmic conformal field theory, log-modular tensor categories and modular forms. J. Phys. Math. Gen. 50:404004. 1605.04630. https://doi.org/10.1088/1751-8121/aa8538 (2017)

  10. Costantino, F., Geer, N., Patureau-Mirand, B.: Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories. J. Topol. 7(4):1005–1053. https://doi.org/10.1112/jtopol/jtu006 (2014)

  11. Costantino, F., Geer, N., Patureau-Mirand, B., Turaev, V.: Kuperberg and Turaev-Viro Invariants in Unimodular Categories, arXiv:1809.07991(2018)

  12. Comes, J., Heidersdorf, T.: Thick ideals in Deligne’s category \(\underline {Rep}(O_{\delta })\). J. Algebra 480:237–265. https://doi.org/10.1016/j.jalgebra.2017.01.050 (2017)

  13. Comes, J., Kujawa, J.R.: Modified traces on Deligne’s category Rep(St). J. Algebraic Combin. 36(4):541–560. https://doi.org/10.1007/s10801-012-0349-1 (2012)

  14. Creutzig, T., Milas, A., Rupert, M.: Logarithmic link invariants of \(\overline {U}_{q}^{H}(\mathfrak {sl}_{2})\) and asymptotic dimensions of singlet vertex algebras. arXiv:1605.05634(2016)

  15. Comes, J.: Ideals in Deligne’s category \(\underline {Rep}(GL_{\delta })\). Math. Res. Lett. 21(5):969–984. https://doi.org/10.4310/MRL.2014.v21.n5.a4 (2014)

  16. Costello, K.: Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210(1):165–214. https://doi.org/10.1016/j.aim.2006.06.004 (2007)

  17. Chari, V., Pressley, A: A guide to quantum groups. Cambridge University Press, Cambridge (1994)

  18. De Renzi, M., Geer, N., Patureau-Mirand, B.: Renormalized Hennings Invariants and 2 + 1-TQFTs. arXiv:1707.08044 (2017)

  19. Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, volume 205 of Mathematical Surveys and Monographs. American Mathematical Society, Providence. https://doi.org/10.1090/surv/205 (2015)

  20. Fontalvo Orozco, A.F., Gainutdinov, A.M.: Module traces and Hopf group-coalgebras. arXiv:1809.01122 (2018)

  21. Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. In: Proceedings of the School and Workshop on Logarithmic Conformal Field Theory and its Applications (Tehran, 2001), vol. 18, pp. 4593–4638. https://doi.org/10.1142/S0217751X03016860 (2003)

  22. Geer, N., Kujawa, J., Patureau-Mirand, B.: Generalized trace and modified dimension functions on ribbon categories. Selecta Math. (N.S.) 17(2):453–504. https://doi.org/10.1007/s00029-010-0046-7 (2011)

  23. Geer, N., Kujawa, J., Patureau-Mirand, B.: Ambidextrous objects and trace functions for nonsemisimple categories. Proc. Amer. Math. Soc. 141(9):2963–2978. https://doi.org/10.1090/S0002-9939-2013-11563-7 (2013)

  24. Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-normalized link invariants. Compos. Math. 145(1):196–212. https://doi.org/10.1112/S0010437X08003795 (2009)

  25. Geer, N., Patureau-Mirand, B., Virelizier, A.: Traces on ideals in pivotal categories. Quantum Topol. 4(1):91–124. https://doi.org/10.4171/QT/36 (2013)

  26. Gainutdinov, A.M., Runkel, I.: Projective objects and the modified trace in factorisable finite tensor categories. arXiv:1703.00150 (2017)

  27. Jantzen, J.C.: Lectures on quantum groups, volume 6 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1996)

  28. Jantzen, J.C.: Representations of algebraic groups, volume 107 of Mathematical Surveys and Monographs, 2nd edn.. American Mathematical Society, Providence (2003)

  29. Kassel, C.: Quantum groups, volume 155 of Graduate Texts in Mathematics. Springer, New York. https://doi.org/10.1007/978-1-4612-0783-2 (1995)

  30. Keller, B.: Calabi-Yau triangulated categories. In: Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pp. 467–489. Eur. Math. Soc., Zürich. https://doi.org/10.4171/062-1/11 (2008)

  31. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211(1):123–151. https://doi.org/10.1016/j.aim.2006.07.013 (2007)

  32. Kontsevich, M., Soibelman, Y.: Motivic Donaldson-Thomas invariants: summary of results. In: Mirror symmetry and tropical geometry, volume 527 of Contemp. Math., pp. 55–89. Amer. Math. Soc., Providence. https://doi.org/10.1090/conm/527/10400 (2010)

  33. Lentner, S.D.: The unrolled quantum group inside Lusztig’s quantum group of divided powers. arXiv:1702.05164 (2017)

  34. Murakami, J.: Generalized Kashaev invariants for knots in three manifolds. Quantum Topol. 8(1):35–73. https://doi.org/10.4171/QT/86 (2017)

  35. Ng, S.-H., Schauenburg, P.: Higher Frobenius-Schur indicators for pivotal categories. In: Hopf algebras and generalizations, volume 441 of Contemp. Math., pp. 63–90. Amer. Math. Soc., Providence. https://doi.org/10.1090/conm/441/08500 (2007)

  36. Phu Ha, N.: Topological invariants from quantum group \({{\mathscr{U}}}_{\zeta }\mathfrak {sl}(2|1)\) at roots of unity. arXiv:1607.03728 (2016)

  37. Rupert, M.: Logarithmic Hopf link invariants for the unrolled restricted quantum group of sl(2). Master’s thesis, The school of the thesis, The address of the publisher, 7 2016

  38. Shimizu, K.: On unimodular finite tensor categories. arXiv:1402.3482 (2014)

  39. Shimizu, K.: Integrals for finite tensor categories. arXiv:1702.02425 (2017)

  40. Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \({\mathscr{W}}_{p}\) triplet algebra. J. Phys. A 46(44):445203, 40. https://doi.org/10.1088/1751-8113/46/44/445203 (2013)

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Acknowledgements

The authors are pleased to thank the anonymous referee for helpful comments.

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Authors and Affiliations

  1. Mathematics & Statistics, Utah State University, Logan, Utah, 84322, USA

    Nathan Geer

  2. Mathematics Department, University of Oklahoma, Norman, OK, 73019, USA

    Jonathan Kujawa

  3. LMBA, CNRS UMR 6205, Université de Bretagne-Sud, F-56000, Vannes, France

    Bertrand Patureau-Mirand

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  1. Nathan Geer
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  2. Jonathan Kujawa
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  3. Bertrand Patureau-Mirand
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Correspondence to Jonathan Kujawa.

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Presented by: Michela Varagnolo

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Research of the second author was partially supported by a Simons Foundation Collaboration Grant.

Research of the first author has been partially supported by the NSF grants DMS-1452093 and DMS-1664387.

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Geer, N., Kujawa, J. & Patureau-Mirand, B. M-traces in (Non-Unimodular) Pivotal Categories. Algebr Represent Theor 25, 759–776 (2022). https://doi.org/10.1007/s10468-021-10044-y

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