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Trace as an alternative decategorification functor. (English) Zbl 1331.81153
A (de)categorification of a given structure needs not to be unique. In this survey paper, the authors deal with the comparison of Grothendieck decategorification and trace decategorification, both defined by certain decategorification functors: \(K_0\), and respectively \(\text{Tr}\). These two decategorification procedures need not agree in general, and in this sense, a large class of examples is provided. Nevertheless, the main result of the paper, that appears also in [the first author et al., “Current algebras and categorified quantum groups”, Preprint, arXiv:1412.1417] and [M. Khovanov and the fourth author, Quantum Topol. 1, No. 1, 1–92 (2010; Zbl 1206.17015); erratum ibid. 2, No. 1, 97–99 (2011)], asserts that certain \(2\)-categories \(\mathcal{U}(\mathfrak{g})\) simultaneously categorify the quantum group \(\mathbf{U}(\mathfrak{sl}_n)\) via the Grothendieck group decategorification functor \(K_0\) and the trace decategorification functor \(\text{Tr}\).

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B35 Universal enveloping (super)algebras
19A99 Grothendieck groups and \(K_0\)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
13D15 Grothendieck groups, \(K\)-theory and commutative rings
Citations:
Zbl 1206.17015
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[1] Al-Nofayee, A, Simple objects in the heart of a t-structure, J. Pure Appl. Algebra, 213, 54-59, (2009) · Zbl 1156.18004
[2] Baez, J; Dolan, J, Categorification. in higher category theory (evanston, IL, 1997), contemp, Math. Am. Math. Soc. Providence, RI, 230, 1-36, (1998)
[3] Balagovic, M.: Degeneration of trigonometric dynamical difference equations for quantum loop algebras to trigonometric Casimir equations for Yangians (2013). arXiv:1308.2347 · Zbl 1393.17026
[4] Bar-Natan, D, Khovanov’s homology for tangles and cobordisms, Geom. Topol., 9, 1443-1499, (2005) · Zbl 1084.57011
[5] Bar-Natan, D; Morrison, S, The karoubi envelope and lee’s degeneration of Khovanov homology, Algebr. Geom. Topol., 6, 1459-1469, (2006) · Zbl 1130.57012
[6] Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque. Soc. Math. France 100, 5-171 · Zbl 1195.57024
[7] Beliakova, A., Habiro, K., Lauda, A., Webster, B.: Current algebras and categorified quantum groups. In preparation (2014) · Zbl 1407.17027
[8] Beliakova, A., Habiro, K., Lauda, A., Zivkovic, M.: Trace decategorification of categorified quantum sl(2). arXiv:1404.1806 (2014) · Zbl 1361.81075
[9] Blanchet, C, An oriented model for Khovanov homology, J. Knot. Theory Ramifications, 19, 291-312, (2010) · Zbl 1195.57024
[10] Borceux, F.: Handbook of Categorical Algebra. 1. Encyclopedia of Mathematics and its Applications, vol. 50. Cambridge University Press, Cambridge (1994) · Zbl 0803.18001
[11] Brichard, J.: The Center of the Nilcoxeter and 0-Hecke Algebras (2008). arXiv:0811.2590
[12] Brundan, J, Symmetric functions, parabolic category \(\mathcal{O}\)O, and the Springer fiber, Duke Math. J, 143, 41-79, (2008) · Zbl 1145.20003
[13] Brundan, J; Kleshchev, A, Blocks of cyclotomic Hecke algebras and Khovanov-lauda algebras, Invent. Math., 178, 451-484, (2009) · Zbl 1201.20004
[14] Brundan, J; Kleshchev, A, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math., 222, 1883-1942, (2009) · Zbl 1241.20003
[15] Brundan, J; Ostrik, V, Cohomology of spaltenstein varieties, Transform. Groups, 16, 619-648, (2011) · Zbl 1230.14078
[16] Cȧldȧraru, A; Willerton, S, The Mukai pairing. I. A categorical approach, New York J. Math., 16, 61-98, (2010) · Zbl 1214.14013
[17] Cautis, S.: Clasp technology to knot homology via the affine Grassmannian. arXiv:1207.2074 (2012) · Zbl 1356.57013
[18] Cautis, S., Kamnitzer, J., Morrison, S.: Webs and quantum skew Howe duality. arXiv:1210.6437 (2012) · Zbl 1387.17027
[19] Cautis, S., Lauda, A.D.: Implicit structure in 2-representations of quantum groups. Selecta Mathematica (2014). arXiv:1111.1431 · Zbl 1370.17017
[20] Cockett, JRB; Koslowski, J; Seely, RAG, Introduction to linear bicategories. the lambek festschrift: mathematical structures in computer science (Montreal, QC, 1997), Math. Structures Comput. Sci., 10, 165-203, (2000) · Zbl 0991.18007
[21] Cooper, B; Krushkal, V, Handle slides and localizations of categories, Int. Math. Res. Not. IMRN, 10, 2179-2202, (2013) · Zbl 1325.57009
[22] Ganter, N; Kapranov, M, Representation and character theory in 2-categories, Adv. Math., 217, 2268-2300, (2008) · Zbl 1136.18001
[23] Garland, H, The arithmetic theory of loop algebras, J. Algebra, 53, 480-551, (1978) · Zbl 0383.17012
[24] Geer, N; Patureau-Mirand, B; Virelizier, A, Traces on ideals in pivotal categories, Quantum Topol., 4, 91-124, (2013) · Zbl 1275.18017
[25] Ginzburg, V, Lagrangian construction of the enveloping algebra U(sln), C. R. Acad. Sci. Paris Sér. I Math., 312, 907-912, (1991) · Zbl 0749.17009
[26] Joyal, A; Street, R, The geometry of tensor calculus, I. Adv. Math., 88, 55-112, (1991) · Zbl 0738.18005
[27] Kashiwara, M, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., 63, 465-516, (1991) · Zbl 0739.17005
[28] Kashiwara, M, Global crystal bases of quantum groups, Duke Math. J, 69, 455-485, (1993) · Zbl 0774.17018
[29] Kauffman, LH, Spin networks and knot polynomials, Internat. J. Modern Phys. A, 5, 93-115, (1990) · Zbl 0708.57003
[30] Kazhdan, D; Lusztig, G, A topological approach to springer’s representations, Adv. Math., 38, 222-228, (1980) · Zbl 0458.20035
[31] Khovanov, M, A categorification of the Jones polynomial, Duke Math. J, 101, 359-426, (2000) · Zbl 0960.57005
[32] Khovanov, M, A functor-valued invariant of tangles, Algebr. Geom. Topol., 2, 665-741, (2002) · Zbl 1002.57006
[33] Khovanov, M; Lauda, A, A diagrammatic approach to categorification of quantum groups III, Quantum Topol., 1, 1-92, (2010) · Zbl 1206.17015
[34] Khovanov, M., Lauda, A., Mackaay, M., Stošić, M.: Extended graphical calculus for categorified quantum sl(2). Memoirs of the AMS 219. arXiv:1006.2866 (2012) · Zbl 1292.17013
[35] Khovanov, M; Mazorchuk, V; Stroppel, C, A brief review of abelian categorifications, Theory Appl. Categ., 22, 479-508, (2009) · Zbl 1181.18005
[36] Kuperberg, G, Spiders for rank 2 Lie algebras, Comm. Math. Phys., 180, 109-151, (1996) · Zbl 0870.17005
[37] Lauda, AD, A categorification of quantum sl(2), Adv. Math., 225, 3327-3424, (2008) · Zbl 1219.17012
[38] Lauda, AD, An introduction to diagrammatic algebra and categorified quantum \({\mathfrak{sl}}_{2}\)𝔰𝔩2, Bulletin Inst. Math. Academia Sinica, 7, 165-270, (2012) · Zbl 1280.81073
[39] Lauda, AD; Vazirani, M, Crystals from categorified quantum groups, Adv. Math., 228, 803-861, (2011) · Zbl 1246.17017
[40] Lauda, A.D., Queffelec, H., Rose, D.: Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m). arXiv:1212.6076(2012) · Zbl 0232.18009
[41] Loday, J.L.: Cyclic Homology. Fundamental Principles of Mathematical Sciences, vol. 301. Springer-Verlag, Berlin (1998) · Zbl 0885.18007
[42] Lusztig, G, Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., 3, 447-498, (1990) · Zbl 0703.17008
[43] Lusztig, G, Canonical bases arising from quantized enveloping algebras. II. common trends in mathematics and quantum field theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl., 102, 175-201, (1991)
[44] Lusztig, G, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A., 89, 8177-8179, (1992) · Zbl 0760.17011
[45] Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Birkhäuser Boston Inc., Boston (1993) · Zbl 0788.17010
[46] Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1979)
[47] Mackaay, M, Spherical 2-categories and 4-manifold invariants, Adv. Math., 143, 288-348, (1999) · Zbl 0946.57021
[48] Mackaay, M.: sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k). arXiv:0905.2059 (2009) · Zbl 1145.20003
[49] Milnor, J.: Introduction to algebraic \(K\)-theory. Ann. Math. Stud. No. 72, pp 1-184. Princeton University Press, Princeton (1971) · Zbl 0237.18005
[50] Mitchell, B, Rings with several objects, Adv. Math., 8, 1-161, (1972) · Zbl 0232.18009
[51] Morrison, S.: A diagrammatic category for the representation theory of (Uqsln). Thesis (Ph.D.)-University of California, Berkeley, ProQuest LLC, Ann Arbor, MI (2007)
[52] Müger, M, From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Algebra, 180, 81-157, (2003) · Zbl 1033.18002
[53] Penrose, R.: Applications of negative dimensional tensors. Combinat. Math. Appl., Proc. Conf., Oxford, 1969, pp 221-244. Academic Press, London (1971) · Zbl 0216.43502
[54] Queffelec, H., Rose, D.E.V.: The \(s\)\(l\)(\(n\)) foam category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality. arXiv:1405.5920 (2014) · Zbl 1360.57025
[55] Reshetikhin, N; Turaev, V, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys., 127, 1-26, (1990) · Zbl 0768.57003
[56] Rosenberg, J.: Algebraic \(K\)-theory and its Applications. Graduate Texts in Mathematics. Springer-Verlag, New York (1994)
[57] Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 (2008) · Zbl 1213.20007
[58] Savage, A.: Introduction to categorification. arXiv:1401.6037(2014) · Zbl 1195.57024
[59] Street, R.: Low-dimensional topology and higher-order categories. Proceedings of CT95, Halifax, July 9-15 (1995) · Zbl 1084.57011
[60] Turaev, V.G.: Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Math., 18, p 588. Water de Gruyter, Berlin (1994) · Zbl 0812.57003
[61] Varagnolo, M; Vasserot, E, Canonical bases and KLR-algebras, J. Reine Angew. Math., 659, 67-100, (2011) · Zbl 1229.17019
[62] Webster, B.: Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products. arXiv:1001.2020(2010)
[63] Webster, B.: Canonical bases and higher representation theory. arXiv:1209.0051 (2012) · Zbl 1393.17029
[64] Webster, B.: Knot invariants and higher representation theory. arXiv:1309.3796 (2013) · Zbl 1446.57001
[65] Zivkovic, M.: Trace decategorification of categorified quantum \(\mathfrak{sl}_{3}\). arXiv:1405.2314 (2014) · Zbl 1361.81075
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