Quinn’s formula and abelian 3-cocycles for quadratic forms. (English) Zbl 1509.18019
Summary: In pointed braided fusion categories knowing the self-symmetry braiding of simples is theoretically enough to reconstruct the associator and braiding on the entire category (up to twisting by a braided monoidal auto-equivalence). We address the problem to provide explicit associator formulas given only such input. This problem was solved by Quinn in the case of finitely many simples. We reprove and generalize this in various ways. In particular, we show that extra symmetries of Quinn’s associator can still be arranged to hold in situations where one has infinitely many isoclasses of simples.
MSC:
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 19D23 | Symmetric monoidal categories |
| 20J99 | Connections of group theory with homological algebra and category theory |
References:
| [1] | Baues, H., Combinatorial homotopy and 4-dimensional complexes, De Gruyter Expositions in Mathematics, vol. 2 (1991), Berlin: Walter de Gruyter & Co.,, Berlin · Zbl 0716.55001 · doi:10.1515/9783110854480 |
| [2] | Bulacu, D., Caenepeel, S., Torrecillas, B.: The braided monoidal structures on the category of vector spaces graded by the Klein group, Proc. Edinb. Math. Soc. (2) 54(3), 613-641. MR 2837470 (2011) · Zbl 1262.16027 |
| [3] | Braunling, O.: Braided categorical groups and strictifying associators. Homol. Homotopy Appl. 22(2), 295-321 MR 4098945 (2020) · Zbl 1440.18034 |
| [4] | Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories. I, Selecta Math. (N.S.) 16(1), 1-119. MR 2609644 (2010) · Zbl 1201.18005 |
| [5] | Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V., Tensor categories, Mathematical Surveys and Monographs, vol. 205 (2015), Providence: American Mathematical Society, Providence · Zbl 1365.18001 |
| [6] | Eilenberg, S., Mac Lane, S.: On the groups H(π,n). I, Ann. of Math. (2) 58, 55-106. MR 0056295 (1953) · Zbl 0050.39304 |
| [7] | Flanders, H.: Tensor and exterior powers. J. Algebra 7, 1-24. MR 212044 (1967) · Zbl 0159.03701 |
| [8] | Huang, H.-L., Liu, G., Ye, Y.: The braided monoidal structures on a class of linear Gr-categories. Algebr. Represent. Theory 17(4), 1249-1265. MR 3228486 (2014) · Zbl 1305.18022 |
| [9] | Huang, H.-L., Liu, G., Yang, Y., Ye, Y.: Finite quasi-quantum groups of diagonal type, J. Reine Angew. Math. 759, 201-243. MR 4058179 (2020) · Zbl 1453.16036 |
| [10] | Huang, H.-L, Wan, Z, Ye, Y: Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants. Proc. Roy. Soc. Edinburgh Sect. A 150(4), 1937-1964. MR 4122441 (2020) · Zbl 1478.18021 |
| [11] | Johnson, N., Osorno, A.: Modeling stable one-types, Theory Appl. Categ. 26(20), 520-537. MR 2981952 (2012) · Zbl 1252.18034 |
| [12] | Joyal, A., Street, R.: Braided monoidal categories. Macquarie Mathematical Reports, no 860081 (1986) · Zbl 0817.18007 |
| [13] | Joyal, A.: Braided tensor categories. Adv. Math. 102(1), 20-78. MR 1250465 (1993) · Zbl 0817.18007 |
| [14] | Kapustin, A., Saulina, N.: Topological boundary conditions in abelian Chern-Simons theory. Nuclear Phys. B 845(3), 393-435. MR 2755172 (2011) · Zbl 1207.81060 |
| [15] | Mac Lane, S.: Cohomology theory of Abelian groups. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 8-14. MR 0045115 (1950) · Zbl 0049.01401 |
| [16] | Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin. MR 2392026 (2008) · Zbl 1136.11001 |
| [17] | Quinn, F.: Group categories and their field theories. Proceedings of the Kirbyfest, Berkeley. Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 407-453. MR 1734419 (1998) · Zbl 0964.18003 |
| [18] | Sính, H.X.: Gr-catégories (thesis, handwritten manuscript). Université, Paris 7. https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html (1975) |
| [19] | Whitehead, J.H.C.: A certain exact sequencex. Ann. Math. (2) 52, 51-110. MR 35997 (1950) · Zbl 0037.26101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
