Stable maps, Q-operators and category $\mathcal {O}$
HTML articles powered by AMS MathViewer
- by David Hernandez PDF
- Represent. Theory 26 (2022), 179-210 Request permission
Abstract:
Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category $\mathcal {O}$ of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new $R$-matrices in the category $\mathcal {O}$ and we establish that a large family of simple modules, including the prefundamental representations associated to $Q$-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified $QQ^*$-systems in terms of the $R$-matrices we construct.References
- Jonathan Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), no. 3, 555–568. MR 1301623, DOI 10.1007/BF02099423
- Léa Bittmann, Quantum Grothendieck rings as quantum cluster algebras, J. Lond. Math. Soc. (2) 103 (2021), no. 1, 161–197. MR 4203046, DOI 10.1112/jlms.12369
- H. Boos, M. Jimbo, T. Miwa, F. Smirnov, and Y. Takeyama, Hidden Grassmann structure in the $XXZ$ model. II. Creation operators, Comm. Math. Phys. 286 (2009), no. 3, 875–932. MR 2472021, DOI 10.1007/s00220-008-0617-z
- Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), no. 2, 261–283. MR 1137064, DOI 10.1007/BF02102063
- Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), no. 2, 295–326. MR 1405590, DOI 10.2140/pjm.1996.174.295
- Ilaria Damiani, La $R$-matrice pour les algèbres quantiques de type affine non tordu, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 493–523 (French, with English and French summaries). MR 1634087, DOI 10.1016/S0012-9593(98)80104-3
- V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
- B. Enriquez, S. Khoroshkin, and S. Pakuliak, Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), no. 3, 691–725. MR 2350435, DOI 10.1007/s00220-007-0351-y
- Edward Frenkel and David Hernandez, Baxter’s relations and spectra of quantum integrable models, Duke Math. J. 164 (2015), no. 12, 2407–2460. MR 3397389, DOI 10.1215/00127094-3146282
- Igor B. Frenkel and Nai Huan Jing, Vertex representations of quantum affine algebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 24, 9373–9377. MR 973376, DOI 10.1073/pnas.85.24.9373
- Boris Feigin, Michio Jimbo, Tetsuji Miwa, and Eugene Mukhin, Finite type modules and Bethe Ansatz equations, Ann. Henri Poincaré 18 (2017), no. 8, 2543–2579. MR 3671544, DOI 10.1007/s00023-017-0577-y
- Edward Frenkel and Evgeny Mukhin, Combinatorics of $q$-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), no. 1, 23–57. MR 1810773, DOI 10.1007/s002200000323
- Edward Frenkel and Nicolai Reshetikhin, The $q$-characters of representations of quantum affine algebras and deformations of $\scr W$-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998) Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205. MR 1745260, DOI 10.1090/conm/248/03823
- Michael Finkelberg and Alexander Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras, Representations and nilpotent orbits of Lie algebraic systems, Progr. Math., vol. 330, Birkhäuser/Springer, Cham, 2019, pp. 133–304. MR 3971731, DOI 10.1007/978-3-030-23531-4_{6}
- Sachin Gautam and Valerio Toledano Laredo, Meromorphic tensor equivalence for Yangians and quantum loop algebras, Publ. Math. Inst. Hautes Études Sci. 125 (2017), 267–337. MR 3668651, DOI 10.1007/s10240-017-0089-9
- David Hernandez, Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), no. 2, 163–200. MR 2195598, DOI 10.1007/s00031-005-1005-9
- David Hernandez, Simple tensor products, Invent. Math. 181 (2010), no. 3, 649–675. MR 2660455, DOI 10.1007/s00222-010-0256-9
- David Hernandez, Avancées concernant les $R$-matrices et leurs applications [d’après Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,…], Astérisque 407 (2019), Exp. No. 1129, 297–331 (French). Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120–1135. MR 3939280, DOI 10.24033/ast
- David Hernandez, Cyclicity and $R$-matrices, Selecta Math. (N.S.) 25 (2019), no. 2, Paper No. 19, 24. MR 3916087, DOI 10.1007/s00029-019-0465-z
- D. Hernandez, Representations of shifted quantum affine algebras, Preprint arXiv:2010.06996, 2021.
- David Hernandez and Michio Jimbo, Asymptotic representations and Drinfeld rational fractions, Compos. Math. 148 (2012), no. 5, 1593–1623. MR 2982441, DOI 10.1112/S0010437X12000267
- David Hernandez and Bernard Leclerc, Cluster algebras and category $\mathcal {O}$ for representations of Borel subalgebras of quantum affine algebras, Algebra Number Theory 10 (2016), no. 9, 2015–2052. MR 3576119, DOI 10.2140/ant.2016.10.2015
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Masaki Kashiwara, Crystal bases and categorifications—Chern Medal lecture, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 249–258. MR 3966729
- Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math. 211 (2018), no. 2, 591–685. MR 3748315, DOI 10.1007/s00222-017-0754-0
- Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349–426. MR 3758148, DOI 10.1090/jams/895
- S. M. Khoroshkin and V. N. Tolstoy, Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617. MR 1134942, DOI 10.1007/BF02102819
- Bernard Leclerc, Cluster algebras and representation theory, Proceedings of the International Congress of Mathematicians. Volume IV, Hindustan Book Agency, New Delhi, 2010, pp. 2471–2488. MR 2827980
- Davesh Maulik and Andrei Okounkov, Quantum groups and quantum cohomology, Astérisque 408 (2019), ix+209 (English, with English and French summaries). MR 3951025, DOI 10.24033/ast
- Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, DOI 10.1090/S0894-0347-00-00353-2
- Hiraku Nakajima, Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), no. 3, 1057–1097. MR 2144973, DOI 10.4007/annals.2004.160.1057
- Hiraku Nakajima, Quiver varieties and tensor products, Invent. Math. 146 (2001), no. 2, 399–449. MR 1865400, DOI 10.1007/PL00005810
- Andrei Okounkov, On the crossroads of enumerative geometry and geometric representation theory, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 839–867. MR 3966746
- A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, Preprint arXiv:1602.09007
- Petr P. Pushkar, Andrey V. Smirnov, and Anton M. Zeitlin, Baxter $Q$-operator from quantum $K$-theory, Adv. Math. 360 (2020), 106919, 63. MR 4035952, DOI 10.1016/j.aim.2019.106919
- M. Varagnolo and E. Vasserot, Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), no. 3, 509–533. MR 1885830, DOI 10.1215/S0012-7094-02-11135-1
Additional Information
- David Hernandez
- Affiliation: Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, IUF, F-75006 Paris, France
- MR Author ID: 707094
- Email: david.hernandez@u-paris.fr
- Received by editor(s): April 20, 2021
- Received by editor(s) in revised form: September 17, 2021
- Published electronically: March 17, 2022
- Additional Notes: The author was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory 26 (2022), 179-210
- MSC (2020): Primary 17B37, 17B10, 82B23
- DOI: https://doi.org/10.1090/ert/604
- MathSciNet review: 4396040