Monoidal supercategories. (English) Zbl 1396.17012
By now, there are several notions in the literature which are supposed to describe what should be called a super monoidal category. The goal of the paper under review is to clarify these notions and connections between them and in this way to develop a unified axiomatization for \(2\)-supercategories, that is \(2\)-categories enriched in vector superspaces, and related formal constructions. In particular, the paper discusses a superanalogue of the notion of Drinfeld center and considers \(2\)-supercategories with a fixed distinguished involution in its Drinfeld center. Gradings and Grothendieck groups in the supersetting are also discussed.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |
Keywords:
superalgebra; supercategory; superfunctor; 2-supercategory; superadditive envelope; monoidal supercategoryCitations:
Zbl 1391.17011References:
| [1] | Bakalov, B., Kirillov, A., Jr.: Lectures on Tensor Categories and Modular Functors. American Mathematical Society (2001) · Zbl 0965.18002 |
| [2] | Brundan J.: On the definition of Kac-Moody 2-category. Math. Ann. 364, 353-372 (2016) · Zbl 1395.17014 · doi:10.1007/s00208-015-1207-y |
| [3] | Brundan, J., Comes, J., Nash, D., Reynolds, A.: A basis theorem for the oriented Brauer category and its cyclotomic quotients. Quantum Top. (To appear) · Zbl 1419.18011 |
| [4] | Brundan, J., Ellis, A.: Super Kac-Moody 2-categories. arXiv:1701.04133 · Zbl 1442.17008 |
| [5] | Brundan, J., Losev, I., Webster, B.: Tensor product categorifications and the super Kazhdan-Lusztig conjecture. Int. Math. Res. Notices (2016), article ID rnv388, 81 p · Zbl 1405.17045 |
| [6] | Cautis S., Kamnitzer J., Morrison S.: Webs and quantum skew Howe duality. Math. Ann. 360, 351-390 (2014) · Zbl 1387.17027 · doi:10.1007/s00208-013-0984-4 |
| [7] | Clark S.: Quantum supergroups IV: the modified form. Math. Z. 278, 493-528 (2014) · Zbl 1364.17015 · doi:10.1007/s00209-014-1324-4 |
| [8] | Clark S., Wang W.: Canonical basis for quantum \[{{\mathfrak{osp}}(1|2)}\] osp(1|2). Lett. Math. Phys. 103, 207-231 (2013) · Zbl 1293.17016 · doi:10.1007/s11005-012-0592-3 |
| [9] | Elias B., Williamson G.: Soergel calculus. Represent. Theory 20, 295-374 (2016) · Zbl 1427.20006 · doi:10.1090/ert/481 |
| [10] | Elias, B., Williamson, G.: Diagrammatics for Coxeter groups and their braid groups. Quantum Top. (To appear) · Zbl 1383.20021 |
| [11] | Ellis A., Lauda A.: An odd categorification of \[{U_q(\mathfrak{sl}_2)}\] Uq(sl2). Adv. Math. 265, 169-240 (2014) · Zbl 1304.17012 · doi:10.1016/j.aim.2014.07.036 |
| [12] | Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories. American Mathematical Society (2015) · Zbl 1365.18001 |
| [13] | Jung J.H., Kang S.-J.: Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras. J. Algebra 399, 516-545 (2014) · Zbl 1370.17009 · doi:10.1016/j.jalgebra.2013.08.029 |
| [14] | Kang S.-J., Kashiwara M., Oh S.-J.: Supercategorification of quantum Kac-Moody algebras II. Adv. Math. 265, 169-240 (2014) · Zbl 1304.17012 · doi:10.1016/j.aim.2014.07.036 |
| [15] | Kelly, G.M.: Basic concepts of enriched category theory. Reprint of the 1982 original. Repr. Theory Appl. Categ. (10), 1-136 (2005) · Zbl 1086.18001 |
| [16] | Khovanov M., Lauda A.: A categorification of quantum \[{\mathfrak{sl}(n)}\] sl(n). Quantum Top. 1, 1-92 (2010) · Zbl 1206.17015 · doi:10.4171/QT/1 |
| [17] | Kujawa, J., Tharp, B.: The marked Brauer category. J. Lond. Math. Soc. (To appear) · Zbl 1427.17015 |
| [18] | Kuperberg G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180, 109-151 (1996) · Zbl 0870.17005 · doi:10.1007/BF02101184 |
| [19] | Leinster, T.: Basic bicategories. arXiv:math/9810017 · Zbl 1295.18001 |
| [20] | Lehrer G., Zhang R.-B.: The Brauer category and invariant theory. J. Eur. Math. Soc 17, 2311-2351 (2015) · Zbl 1328.14079 · doi:10.4171/JEMS/558 |
| [21] | Mac Lane S.: Categories for the Working Mathematician. Springer, Berlin (1978) · Zbl 0232.18001 · doi:10.1007/978-1-4757-4721-8 |
| [22] | Manin Y.I.: Gauge Field Theory and Complex Geometry. Springer, Berlin (1997) · Zbl 0884.53002 · doi:10.1007/978-3-662-07386-5 |
| [23] | Meir E., Szymik M.: Drinfeld center for bicategories. Doc. Math. 20, 707-735 (2015) · Zbl 1348.18009 |
| [24] | Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 · Zbl 1247.20002 |
| [25] | Soergel W.: Kazhdan-Lusztig-Polynome und unzerlegbare bimoduln über polynomringen. J. Inst. Math. Jussieu 6, 501-525 (2007) · Zbl 1192.20004 · doi:10.1017/S1474748007000023 |
| [26] | Usher, R.: Fermionic 6j-symbols in superfusion categories. arXiv:1606.03466 · Zbl 1397.18019 |
| [27] | Westbury B.: The representation theory of the Temperley-Lieb algebras. Math. Z. 219, 539-565 (1995) · Zbl 0840.16008 · doi:10.1007/BF02572380 |
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.
