A diagrammatic approach to categorification of quantum groups I

Khovanov, Mikhail; Lauda, Aaron

doi:10.1090/S1088-4165-09-00346-X

A diagrammatic approach to categorification of quantum groups I
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by Mikhail Khovanov and Aaron D. Lauda

Represent. Theory 13 (2009), 309-347

DOI: https://doi.org/10.1090/S1088-4165-09-00346-X

Published electronically: July 28, 2009

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Abstract:

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak {g})$, where $\mathfrak {g}$ is the Kac-Moody Lie algebra associated with the graph.

References

Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
Susumu Ariki, Lectures on cyclotomic Hecke algebras, Quantum groups and Lie theory (Durham, 1999) London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 1–22. MR 1903956
S. Ariki. Representations of quantum algebras and combinatorics of Young tableaux, volume 26 of University Lecture Series. AMS, Providence, RI, 2002.
A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak {sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241. MR 1714141, DOI 10.1007/s000290050047
Sara Billey and V. Lakshmibai, Singular loci of Schubert varieties, Progress in Mathematics, vol. 182, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1782635, DOI 10.1007/978-1-4612-1324-6
Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$, Represent. Theory 5 (2001), 317–403. MR 1870595, DOI 10.1090/S1088-4165-01-00123-6
Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and $\mathfrak {sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. MR 2373155, DOI 10.4007/annals.2008.167.245
Louis Crane and Igor B. Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), no. 10, 5136–5154. Topology and physics. MR 1295461, DOI 10.1063/1.530746
Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Igor Frenkel, Mikhail Khovanov, and Catharina Stroppel, A categorification of finite-dimensional irreducible representations of quantum $\mathfrak {sl}_2$ and their tensor products, Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431. MR 2305608, DOI 10.1007/s00029-007-0031-y
I. Grojnowski. Affine $sl_p$ controls the representation theory of the symmetric group and related Hecke algebras, 1999, math.RT/9907129.
I. Grojnowski and G. Lusztig, On bases of irreducible representations of quantum $\textrm {GL}_n$, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 167–174. MR 1197834, DOI 10.1090/conm/139/1197834
I. Grojnowski and G. Lusztig, A comparison of bases of quantized enveloping algebras, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 11–19. MR 1247495, DOI 10.1090/conm/153/01304
I. Grojnowski and M. Vazirani, Strong multiplicity one theorems for affine Hecke algebras of type A, Transform. Groups 6 (2001), no. 2, 143–155. MR 1835669, DOI 10.1007/BF01597133
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
Masaki Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425, DOI 10.1007/BF02097367
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Mikhail Khovanov, Nilcoxeter algebras categorify the Weyl algebra, Comm. Algebra 29 (2001), no. 11, 5033–5052. MR 1856929, DOI 10.1081/AGB-100106800
Alexander S. Kleshchev, Branching rules for modular representations of symmetric groups. II, J. Reine Angew. Math. 459 (1995), 163–212. MR 1319521, DOI 10.1515/crll.1995.459.163
A. S. Kleshchev, Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. (2) 54 (1996), no. 1, 25–38. MR 1395065, DOI 10.1112/jlms/54.1.25
Alexander Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. (3) 75 (1997), no. 3, 497–558. MR 1466660, DOI 10.1112/S0024611597000427
Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR 2165457, DOI 10.1017/CBO9780511542800
Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986), no. 3, 187–237. MR 866159, DOI 10.1016/0001-8708(86)90101-5
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572, DOI 10.1007/BF02101678
A. D. Lauda. A categorification of quantum sl(2), 2008, arXiv:0803.3652.
A. D. Lauda. Categorified quantum sl(2) and equivariant cohomology of iterated flag variaties, 2008, arXiv:0803.3848.
Bernard Leclerc, Dual canonical bases, quantum shuffles and $q$-characters, Math. Z. 246 (2004), no. 4, 691–732. MR 2045836, DOI 10.1007/s00209-003-0609-9
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175–201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, DOI 10.1143/PTPS.102.175
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
G. Lusztig, Tight monomials in quantized enveloping algebras, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 117–132. MR 1261904
L. Manivel. Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 of SMF/AMS Texts and Monographs. AMS, Providence, RI, 2001.
Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
Constantin Năstăsescu and Freddy Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol. 1836, Springer-Verlag, Berlin, 2004. MR 2046303, DOI 10.1007/b94904
A. Postnikov. Enumeration in algebra and geometry. Ph.D. thesis, MIT, 1997 available at http://citeseer.ist.psu.edu/postnikov97enumeration.html.
Markus Reineke, Monomials in canonical bases of quantum groups and quadratic forms, J. Pure Appl. Algebra 157 (2001), no. 2-3, 301–309. MR 1812057, DOI 10.1016/S0022-4049(00)00008-6
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
R. Rouquier. Higher representation theory. Talk at IAS, March 2008.
R. Rouquier. Higher representations of Kac-Moody algebras. Work in progress.
J. Sussan. Category O and sl(k) link invariants, 2007, math.QA/0701045.
M. Vazirani. Irreducible modules over the affine Hecke algebra: a strong multiplicity one result. Ph.D. thesis, UC Berkeley, 1999, math.RT/0107052.
M. Vazirani, Parameterizing Hecke algebra modules: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs, Transform. Groups 7 (2002), no. 3, 267–303. MR 1923974, DOI 10.1007/s00031-002-0014-1
A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084, DOI 10.24033/asens.1379
H. Zheng. A geometric categorification of tensor products of ${U}_q(sl_2)$-modules, 2007, arXiv:0705.2630.
H. Zheng. Categorification of integrable representations of quantum groups, 2008, arXiv:0803.3668.