×

Diagram categories for \(\mathrm{U}_q\)-tilting modules at roots of unity. (English) Zbl 1427.16025

Summary: We give a diagrammatic presentation of the category of \(\mathbf{U}_{q}(\mathfrak{sl}_{2})\)-tilting modules \(\mathfrak T\) for \(q\) being a root of unity and introduce a grading on \(\mathfrak T\). This grading is a “root of unity phenomenon” and might lead to new insights about link and 3-manifold invariants deduced from \(\mathfrak T\). We also give a diagrammatic category for the (graded) projective endofunctors on \(\mathfrak T\), indicate how our results could generalize, and collect some “well-known” facts to give a reasonably self-contained exposition.

MSC:

16T20 Ring-theoretic aspects of quantum groups
16G99 Representation theory of associative rings and algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
18M05 Monoidal categories, symmetric monoidal categories
20G42 Quantum groups (quantized function algebras) and their representations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. H. Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), no. 1, 149-159. · Zbl 0760.17004
[2] H. H. Andersen, The strong linkage principle for quantum groups at roots of 1, J. Algebra 260 (2003), no. 1, 2-15. · Zbl 1043.17005
[3] H. H. Andersen, J. C. Jantzen, W. Soergel, Representations of Quantum Groups at a p-th Root of Unity and of Semisimple Groups in Characteristic p: Independence of p, Astérisque 220 (1994). · Zbl 0802.17009
[4] H. H. Andersen, M. Kaneda, Rigidity of tilting modules, Mosc. Math. J. 11 (2011), no. 1, 1-39. · Zbl 1229.17012
[5] H. H. Andersen, P. Polo, K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1-59. · Zbl 0724.17012
[6] I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras. Vol. 1: Techniques of Representation Theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, 2006. · Zbl 1092.16001
[7] A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. · Zbl 0864.17006
[8] J. N. Bernstein, I. Frenkel, M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199-241. · Zbl 0981.17001
[9] J. N. Bernstein, S. I. Gel’fand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 4, 245-285. · Zbl 0445.17006
[10] T. Braden, Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002), no. 3, 493-532. · Zbl 1009.32019
[11] J. Brundan, C. Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709-773. · Zbl 1326.17006
[12] J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685-722. · Zbl 1275.17012
[13] J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra II: Koszulity, Transform. Groups 15 (2010), no. 1, 1-45. · Zbl 1205.17010
[14] J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra III: category O, Represent. Theory 15 (2011), 170-243. · Zbl 1261.17006
[15] J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. 14 (2012), no. 2, 373-419. · Zbl 1243.17004
[16] Y. Chen, M. Khovanov, An invariant of tangle cobordisms via subquotients of arc rings, Fund. Math. 225 (2014), 23-44. · Zbl 1321.57031
[17] B. Elias, The two-color Soergel calculus, preprint, arXiv:1308.6611 (2013). · Zbl 1382.20006
[18] B. Elias, M. Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. 2010 (2010), Article ID 978635. · Zbl 1219.18003
[19] B. Elias, N. Libedinsky, Soergel bimodules for universal Coxeter groups, preprint, arXiv:1401.2467 (2014). · Zbl 1435.20009
[20] B. Elias, G. Williamson, Diagrammatics for Coxeter groups and their braid groups, preprint, arXiv:1405.4928 (2014). · Zbl 1383.20021
[21] B. Elias, G. Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089-1136. · Zbl 1326.20005
[22] B. Elias, G. Williamson, Soergel calculus, preprint, arXiv:1309.0865 (2013). · Zbl 1427.20006
[23] S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39-60. · Zbl 0798.20035
[24] J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG CategoryO \[\mathcal{O} \], Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008. · Zbl 1177.17001
[25] J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996. · Zbl 0842.17012
[26] J. C. Jantzen, Representations of Algebraic Groups, Mathematical Surveys and Monographs 107, 2nd ed., American Mathematical Society, Providence, RI, 2003. · Zbl 1034.20041
[27] A. Joyal, R. Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991), no. 1, 55-112. · Zbl 0738.18005
[28] M. Kashiwara, T. Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1996), no. 1, 21-62. · Zbl 0829.17020
[29] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras I-IV, J. Amer. Math. Soc. 6 (1994), no. 4, 905-947, 949-1011 and J. Amer. Math. Soc. 7 (1994), no. 2, 335-381, 383-453. · Zbl 0802.17008
[30] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. · Zbl 0960.57005
[31] M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665-741. · Zbl 1002.57006
[32] M. Khovanov, Categorifications from planar diagrammatics, Japanese J. Math. 5 (2010), no. 2, 153-181. · Zbl 1226.81094
[33] M. Khovanov, A. D. Lauda, A categorification of quantumsl \[\mathfrak{s}\mathfrak{l}\] n, Quantum Topol. 2 (2010), no. 1, 1-92. · Zbl 1206.17015
[34] M. Khovanov, A. D. Lauda, M. Mackaay, M. Stošić, Extended Graphical Calculus for Categorified Quantumsl \[\mathfrak{s}\mathfrak{l} 2\], Mem. Amer. Math. Soc., Vol. 219-1029, 2012. · Zbl 1292.17013
[35] M. Khovanov, L. Rozansky, Matrix factorizations and link homology I, Fund. Math. 199 (2008), no. 1, 1-91. · Zbl 1145.57009
[36] M. Khovanov, P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203-271. · Zbl 1035.53122
[37] A. D. Lauda, An introduction to diagrammatic algebra and categorified quantum sl(2), Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 2, 165-270. · Zbl 1280.81073
[38] A. D. Lauda, H. Queffelec, D. E. V. Rose, Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m), Algebr. Geom. Topol. 15 (2015), no. 5, 2517-2608. · Zbl 1330.81128
[39] N. Libedinsky, Presentation of right-angled Soergel categories by generators and relations, J. Pure Appl. Algebra 214 (2010), no. 12, 2265-2278. · Zbl 1252.20002
[40] G. Lusztig, Introduction to Quantum Groups, Modern Birkhäuser Classics, Birkhäuser, Basel, 2010. · Zbl 1246.17018
[41] G. Lusztig, Modular representations and quantum groups, in: Classical Groups and Related Topics (Beijing, 1987), Contemp. Math. 82 (1989), pp. 59-77. · Zbl 0665.20022
[42] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), no. 1-3, 89-113. · Zbl 0714.17013
[43] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Sympos. Pure Math. 37 (1980), 313-317. · Zbl 0453.20005
[44] M. Mackaay, Thesl \[\mathfrak{s}\mathfrak{l} (N)\]-web algebras and dual canonical bases, J. Algebra 409 (2014), 54-100. · Zbl 1368.17019
[45] M. Mackaay, W. Pan, D. Tubbenhauer, Thesl \[\mathfrak{s}\mathfrak{l} 3\]-web algebra, Math. Z. 277 (2014), no. 1-2, 401-479. · Zbl 1321.17010
[46] V. Mazorchuk, S. Ovsienko, C. Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1129-1172. · Zbl 1229.16018
[47] M. Müger, From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Alg. 180 (2003), no. 1-2, 81-157. · Zbl 1033.18002
[48] J. Paradowski, Filtration of modules over the quantum algebra, Proc. Sympos. Pure Math., Vol. 56, Part 2 (1994), 93-108. · Zbl 0831.20059
[49] H. Queffelec, D. E. V. Rose, Thesl \[\mathfrak{s}\mathfrak{l}\] nfoam 2-category: A combinatorial formulation of Khovanov-Rozansky homology via categorical skew-Howe duality, preprint, arXiv:1405.5920 (2014). · Zbl 1360.57025
[50] C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209-223. · Zbl 0725.16011
[51] R. Rouquier, 2-Kac-Moody algebras, preprint, arXiv:0812.5023 (2008).
[52] S. F. Sawin, Quantum groups at roots of unity and modularity, J. Knot Theory Ramifications 15 (2006), no. 10, 1245-1277. · Zbl 1117.17006
[53] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-448 (electronic). · Zbl 0964.17018
[54] W. Soergel, Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421-445. · Zbl 0747.17008
[55] W. Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83-114 (electronic). · Zbl 0886.05123
[56] W. Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49-74. · Zbl 0745.22014
[57] C. Stroppel, Categorification of the Temperley-Lieb category, tangles and cobordisms via projective functors, Duke Math. J. 126 (2005), no. 3, 547-596. · Zbl 1112.17010
[58] C. Stroppel, Category O: Gradings and translation functors, J. Algebra 268 (2003), no. 1, 301-326. · Zbl 1040.17002
[59] C. Stroppel, TQFT with corners and tilting functors in the Kac-Moody case, preprint, arXiv:math/0605103 (2006).
[60] D. Tubbenhauer, sl \[\mathfrak{s}\mathfrak{l} 3\]-web bases, intermediate crystal bases and categorification, J. Algebr. Combin. 40(4) (2014), no. 4, 1001-1076. · Zbl 1322.17010
[61] D. Tubbenhauer, sl \[\mathfrak{s}\mathfrak{l}\] n-webs, categorification and Khovanov-Rozansky homologies, preprint, arXiv:1404.5752 (2014).
[62] V. Turaev, Quantum Invariants of Knots and 3-Manifolds, 2nd revised ed., de Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter, Berlin, 2010. · Zbl 1213.57002
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.