Frobenius Monoidal Functors of Dijkgraaf–Witten Categories and Rigid Frobenius Algebras

Samuel Hannah   ; Robert Laugwitz   ; Ana Ros Camacho

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

We construct a separable Frobenius monoidal functor from \smash{$\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$} to $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$, recovering the classification of étale algebras in these categories by Davydov–Simmons [J. Algebra 471 (2017), 149–175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov–Ostrik [Adv. Math. 171 (2002), 183–227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz–Walton [Comm. Math. Phys., {t}o appear, arXiv:2202.08644] in the general case.

Détail

Keywords: Frobenius monoidal functor, Frobenius algebra, Dijkgraaf–Witten category, local module, modular tensor category
Mots-clés : étale algebra.

Bibliographie

[1] Aguiar M., Mahajan S., Monoidal functors, species and Hopf algebras, CRM Monogr. Ser., 29, American Mathematical Society, Providence, RI, 2010 | DOI | MR | Zbl

[2] Bartlett B., Douglas C.L., Schommer-Pries C.J., Vicary J., Modular categories as representations of the 3-dimensional bordism 2-category, arXiv: math.AT/1509.06811

[3] Benson D.J., Representations and cohomology, v. I, Cambridge Stud. Adv. Math., 30, Basic representation theory of finite groups and associative algebras, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[4] Brown K.S., Cohomology of groups, Grad. Texts Math., 87, Springer, New York, 1982 | DOI | MR | Zbl

[5] Bruguières A., Natale S., “Exact sequences of tensor categories”, Int. Math. Res. Not., 2011 (2011), 5644–5705, arXiv: 1006.0569 | DOI | MR | Zbl

[6] Creutzig T., Kanade S., McRae R., Tensor categories for vertex operator superalgebra extensions, arXiv: 1705.05017

[7] Davydov A., “Modular invariants for group-theoretical modular data. I”, J. Algebra, 323 (2010), 1321–1348, arXiv: 0908.1044 | DOI | MR | Zbl

[8] Davydov A., Müger M., Nikshych D., Ostrik V., “The Witt group of non-degenerate braided fusion categories”, J. Reine Angew. Math., 677 (2013), 135–177, arXiv: 1009.2117 | DOI | MR | Zbl

[9] Davydov A., Simmons D., “On Lagrangian algebras in group-theoretical braided fusion categories”, J. Algebra, 471 (2017), 149–175, arXiv: 1603.04650 | DOI | MR | Zbl

[10] De Renzi M., Gainutdinov A.M., Geer N., Patureau-Mirand B., Runkel I., “3-Dimensional TQFTs from non-semisimple modular categories”, Selecta Math. (N.S.), 28 (2022), 42, 60 pp., arXiv: 1912.02063 | DOI | MR | Zbl

[11] Dijkgraaf R., Pasquier V., Roche P., “Quasi Hopf algebras, group cohomology and orbifold models”, Nuclear Phys. B Proc. Suppl., 18 (1991), 60–72 | DOI | MR

[12] Dijkgraaf R., Pasquier V., Roche P., Topological interactions in broken gauge theories, arXiv: hep-th/9511195

[13] Dong C., Lepowsky J., Generalized vertex algebras and relative vertex operators, Progr. Math., 112, Birkhäuser, Boston, MA, 1993 | DOI | MR | Zbl

[14] Dong C., Ren L., Xu F., “On orbifold theory”, Adv. Math., 321 (2017), 1–30, arXiv: 1507.03306 | DOI | MR | Zbl

[15] Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr., 205, American Mathematical Society, Providence, RI, 2015 | DOI | MR | Zbl

[16] Flake J., Harman N., Laugwitz R., “The indecomposable objects in the center of Deligne's category $\underline{\rm Re}{\rm p}S_t$”, Proc. Lond. Math. Soc., 126 (2023), 1134–1181, arXiv: 2105.10492 | DOI | MR

[17] Fröhlich J., Fuchs J., Runkel I., Schweigert C., “Correspondences of ribbon categories”, Adv. Math., 199 (2006), 192–329, arXiv: math.CT/0309465 | DOI | MR | Zbl

[18] Fuchs J., Runkel I., Schweigert C., “TFT construction of RCFT correlators. I: partition functions”, Nuclear Phys. B, 646 (2002), 353–497, arXiv: hep-th/0204148 | DOI | MR | Zbl

[19] Fuchs J., Schaumann G., Schweigert C., “A modular functor from state sums for finite tensor categories and their bimodules”, Theory Appl. Categ., 38, 2022, 436–594, arXiv: 1911.06214 | MR | Zbl

[20] Fuchs J., Stigner C., “On Frobenius algebras in rigid monoidal categories”, Arab. J. Sci. Eng. Sect. C Theme Issues, 33 (2008), 175–191, arXiv: 0901.4886 | MR | Zbl

[21] Hochschild G., Serre J.-P., “Cohomology of group extensions”, Trans. Amer. Math. Soc., 74 (1953), 110–134 | DOI | MR | Zbl

[22] Huang Y.-Z., “Rigidity and modularity of vertex tensor categories”, Commun. Contemp. Math., 10 (2008), 871–911, arXiv: math.QA/0502533 | DOI | MR | Zbl

[23] Huang Y.Z., Kirillov Jr. A., Lepowsky J., “Braided tensor categories and extens-ions of vertex operator algebras”, Comm. Math. Phys., 337 (2015), 1143–1159, arXiv: 1406.3420 | DOI | MR | Zbl

[24] Huang Y.-Z., Lepowsky J., Zhang L., “Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra”, Conformal Field Theories and Tensor Categories, Math. Lect. Peking Univ., Springer, Heidelberg, 2014, 169–248, arXiv: 1110.1931 | DOI | MR | Zbl

[25] Kerler T., Lyubashenko V.V., Non-semisimple topological quantum field theories for 3-manifolds with corners, Lecture Notes in Math., 1765, Springer, Berlin, 2001 | DOI | MR | Zbl

[26] Kirillov Jr. A., Ostrik V., “On a $q$-Analogue of the McKay correspondence and the ADE classification of $\mathfrak{sl}_2$ conformal field theories”, Adv. Math., 171 (2002), 183–227, arXiv: math.QA/0101219 | DOI | MR | Zbl

[27] Laugwitz R., Walton C., “Constructing non-semisimple modular categories with relative monoidal centers”, Int. Math. Res. Not., 2022 (2022), 15826–15868, arXiv: 2010.11872 | DOI | MR | Zbl

[28] Laugwitz R., Walton C., “Constructing non-semisimple modular categories with local modules”, Comm. Math. Phys. (to appear) , arXiv: 2202.08644 | DOI | MR

[29] Lentner S., Mierach S.N., Schweigert C., Sommerhaeuser Y., Hochschild cohomology, modular tensor categories, and mapping class groups, v. I, SpringerBriefs Math. Phys., 44, Springer, Berlin, 2023 | DOI

[30] Lentner S.D., “Quantum groups and Nichols algebras acting on conformal field theories”, Adv. Math., 378 (2021), 107517, 71 pp., arXiv: 1702.06431 | DOI | MR | Zbl

[31] Majid S., “Quantum double for quasi-Hopf algebras”, Lett. Math. Phys., 45 (1998), 1–9, arXiv: q-alg/9701002 | DOI | MR | Zbl

[32] McRae R., “Twisted modules and $G$-equivariantization in logarithmic conformal field theory”, Comm. Math. Phys., 383 (2021), 1939–2019, arXiv: 1910.13226 | DOI | MR | Zbl

[33] Möller S., A cyclic orbifold theory for holomorphic vertex operator algebras and applications, Ph.D. Thesis, Technische Universität Darmstadt, 2016, arXiv: 1611.09843

[34] Morales Y., Müller M., Plavnik J., Ros Camacho A., Tabiri A., Walton C., “Algebraic structures in group-theoretical fusion categories”, Algebr. Represent. Theory (to appear) , arXiv: 2001.03837 | DOI

[35] Natale S., “On the equivalence of module categories over a group-theoretical fusion category”, SIGMA, 13 (2017), 042, 9 pp., arXiv: 1608.04435 | DOI | MR | Zbl

[36] Ostrik V., “Module categories over the {D}rinfeld double of a finite group”, Int. Math. Res. Not., 2003 (2003), 1507–1520, arXiv: math.QA/0111139 | DOI | MR | Zbl

[37] Pareigis B., “On braiding and dyslexia”, J. Algebra, 171 (1995), 413–425 | DOI | MR | Zbl

[38] Radford D.E., Hopf algebras, Ser. Knots Everything, 49, World Scientific Publishing, Hackensack, NJ, 2012 | DOI | MR | Zbl

[39] Reshetikhin N., Turaev V.G., “Invariants of $3$-manifolds via link polynomials and quantum groups”, Invent. Math., 103 (1991), 547–597 | DOI | MR | Zbl

[40] Schauenburg P., “The monoidal center construction and bimodules”, J. Pure Appl. Algebra, 158 (2001), 325–346 | DOI | MR | Zbl

[41] Schweigert C., Woike L., “Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories”, Adv. Math., 386 (2021), 107814, 55 pp., arXiv: 2004.14343 | DOI | MR | Zbl

[42] Shimizu K., “Non-degeneracy conditions for braided finite tensor categories”, Adv. Math., 355 (2019), 106778, 36 pp., arXiv: 1602.06534 | DOI | MR | Zbl

[43] Shimizu K., “Ribbon structures of the Drinfeld center of a finite tensor category”, Kodai Math. J., 46 (2023), 75–114, arXiv: 1707.09691 | DOI | MR

[44] Turaev V.G., Quantum invariants of knots and 3-manifolds, De Gruyter Stud. Math., 18, De Gruyter, Berlin, 1994 | DOI | MR | Zbl

[45] Weibel C.A., An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Cambridge University Press, Cambridge, 1994 | DOI | MR | Zbl

[46] Willerton S., “The twisted {D}rinfeld double of a finite group via gerbes and finite groupoids”, Algebr. Geom. Topol., 8 (2008), 1419–1457, arXiv: math.QA/0503266 | DOI | MR | Zbl

[47] Yadav H., Frobenius monoidal functors from (co)Hopf adjunctions, arXiv: 2209.15606