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Monoidal categorification of cluster algebras


Authors: Seok-Jin Kang, Masaki Kashiwara, Myungho Kim and Se-jin Oh
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 13F60, 81R50, 16Gxx, 17B37
DOI: https://doi.org/10.1090/jams/895
Published electronically: December 5, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the quantum cluster algebra structure of a
unipotent quantum coordinate ring $ A_q(\mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $ w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda-Rouquier
algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $ R$-modules to become a monoidal categorification, where $ R$ is a symmetric Khovanov-Lauda-Rouquier algebra.
Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $ A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $ q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.


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  • [1] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171 (French). MR 751966
  • [2] Arkady Berenstein and Andrei Zelevinsky, String bases for quantum groups of type $ A_r$, I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 51-89. MR 1237826
  • [3] Arkady Berenstein and Andrei Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405-455. MR 2146350
  • [4] Giovanni Cerulli Irelli, Bernhard Keller, Daniel Labardini-Fragoso, and Pierre-Guy Plamondon, Linear independence of cluster monomials for skew-symmetric cluster algebras, Compos. Math. 149 (2013), no. 10, 1753-1764. MR 3123308
  • [5] Ben Davison, Davesh Maulik, Jörg Schürmann, and Balázs Szendrői, Purity for graded potentials and quantum cluster positivity, Compos. Math. 151 (2015), no. 10, 1913-1944. MR 3414389
  • [6] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642
  • [7] Christof Geiss, Bernard Leclerc, and Jan Schröer, Semicanonical bases and preprojective algebras, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 2, 193-253 (English, with English and French summaries). MR 2144987
  • [8] Christof Geiß, Bernard Leclerc, and Jan Schröer, Kac-Moody groups and cluster algebras, Adv. Math. 228 (2011), no. 1, 329-433. MR 2822235
  • [9] Christof Geiß, Bernard Leclerc, and Jan Schröer, Cluster algebra structures and semicanonical bases for unipotent groups, arXiv:0703039v4 [math.RT].
  • [10] Christof Geiss, Bernard Leclerc, and Jan Schröer, Factorial cluster algebras, Doc. Math. 18 (2013), 249-274. MR 3064982
  • [11] C. Geiß, B. Leclerc, and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), no. 2, 337-397. MR 3090232
  • [12] David Hernandez and Bernard Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), no. 2, 265-341. MR 2682185
  • [13] David Hernandez and Bernard Leclerc, Monoidal categorifications of cluster algebras of type $ A$ and $ D$, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., vol. 40, Springer, Heidelberg, 2013, pp. 175-193. MR 3077685
  • [14] Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-Jin Oh, Symmetric quiver Hecke algebras and $ R$-matrices of quantum affine algebras IV, Selecta Math. (N.S.) 22 (2016), no. 4, 1987-2015. MR 3573951
  • [15] Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Simplicity of heads and socles of tensor products, Compos. Math. 151 (2015), no. 2, 377-396. MR 3314831
  • [16] M. Kashiwara, On crystal bases of the $ Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516. MR 1115118
  • [17] Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455-485. MR 1203234
  • [18] Masaki Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839-858. MR 1240605
  • [19] Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383-413. MR 1262212
  • [20] Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155-197. MR 1357199
  • [21] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. MR 2525917
  • [22] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. MR 2763732
  • [23] Yoshiyuki Kimura, Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52 (2012), no. 2, 277-331. MR 2914878
  • [24] Yoshiyuki Kimura and Fan Qin, Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261-312. MR 3228430
  • [25] Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki, $ T$-systems and $ Y$-systems in integrable systems, J. Phys. A 44 (2011), no. 10, 103001, 146. MR 2773889
  • [26] Philipp Lampe, A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. IMRN 13 (2011), 2970-3005. MR 2817684
  • [27] P. Lampe, Quantum cluster algebras of type $ A$ and the dual canonical basis, Proc. Lond. Math. Soc. (3) 108 (2014), no. 1, 1-43. MR 3162819
  • [28] Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803-861. MR 2822211
  • [29] B. Leclerc, Imaginary vectors in the dual canonical basis of $ U_q(\mathfrak{n})$, Transform. Groups 8 (2003), no. 1, 95-104. MR 1959765
  • [30] Kyungyong Lee and Ralf Schiffler, Positivity for cluster algebras, Ann. of Math. (2) 182 (2015), no. 1, 73-125. MR 3374957
  • [31] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415
  • [32] G. Lusztig, Canonical bases in tensor products, Proc. Natl. Acad. Sci. USA 89 (1992), no. 17, 8177-8179. MR 1180036
  • [33] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
  • [34] Peter J. McNamara, Representations of Khovanov-Lauda-Rouquier algebras III: symmetric affine type, Math. Z. 287 (2017), no. 1-2, 243-286. MR 3694676
  • [35] Hiraku Nakajima, Cluster algebras and singular supports of perverse sheaves, Advances in representation theory of algebras, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2013, pp. 211-230. MR 3220538
  • [36] Hiraku Nakajima, Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), no. 1, 71-126. MR 2784748
  • [37] Fan Qin, Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J. 166 (2017), no. 12, 2337-2442. MR 3694569
  • [38] R. Rouquier, 2-Kac-Moody algebras, arXiv:0812.5023v1.
  • [39] Raphaël Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359-410. MR 2908731
  • [40] M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67-100. MR 2837011

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Additional Information

Seok-Jin Kang
Affiliation: Research Institute of Computers, Information and Communication, Pusan National University, 2, Busandaehak-ro Pusan 46241, Korea
Email: soccerkang@hotmail.com

Masaki Kashiwara
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email: masaki@kurims.kyoto-u.ac.jp

Myungho Kim
Affiliation: Department of Mathematics, Kyung Hee University, Seoul 02447, Korea
Email: mkim@khu.ac.kr

Se-jin Oh
Affiliation: Department of Mathematics Ewha Womans University, Seoul 03760, Korea
Email: sejin092@gmail.com

DOI: https://doi.org/10.1090/jams/895
Keywords: Cluster algebra, quantum cluster algebra, monoidal categorification, Khovanov--Lauda--Rouquier algebra, unipotent quantum coordinate ring, quantum affine algebra
Received by editor(s): February 15, 2015
Received by editor(s) in revised form: December 19, 2016, and July 15, 2017
Published electronically: December 5, 2017
Additional Notes: This work was supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2017R1C1B2007824).
This work was supported by NRF Grant # 2016R1C1B2013135.
This research was supported by Ministry of Culture, Sports and Tourism (MCST) and Korea Creative Content Agency (KOCCA) in the Culture Technology (CT) Research & Development Program 2017.
Article copyright: © Copyright 2017 American Mathematical Society

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