Multiplicative slices, relativistic Toda and shifted quantum affine algebras. (English) Zbl 1436.17021
Gorelik, Maria (ed.) et al., Representations and nilpotent orbits of Lie algebraic systems. In honour of the 75th birthday of Tony Joseph. Based on the 2017 conferences “Algebraic Modes of Representations”, Weizmann Institute of Science, Rehovot, Israel, July 16–18, 2017 and University of Haifa, Haifa, Israel, July 19–13, 2017. Cham: Birkhäuser. Prog. Math. 330, 133-304 (2019).
Summary: We introduce the shifted quantum affine algebras. They map homomorphically into the quantized \(K\)-theoretic Coulomb branches of \(3d\mathcal{N}=4\) SUSY quiver gauge theories. In type \(A\), they are endowed with a coproduct, and they act on the equivariant \(K\)-theory of parabolic Laumon spaces. In type \(A_1\), they are closely related to the type \(A\) open relativistic quantum Toda system.
For the entire collection see [Zbl 1432.17001].
For the entire collection see [Zbl 1432.17001].
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
References:
| [1] | M. Atiyah, N. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton University Press (1988). · Zbl 0671.53001 |
| [2] | A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445-495. · Zbl 0948.53045 · doi:10.4310/jdg/1214460860 |
| [3] | A. Beilinson, Coherent sheaves on \({\mathbb{P}}^n\) and problems in linear algebra, Funct. Anal. Appl. 12 (1978), no. 3, 214-216. · Zbl 0424.14003 · doi:10.1007/BF01681436 |
| [4] | M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn, H. Kim, Vortices and Vermas, Adv. Theor. Math. Phys. 22 (2018), no. 4, 803-917. · Zbl 07430942 · doi:10.4310/ATMP.2018.v22.n4.a1 |
| [5] | V. Baranovsky, S. Evens, V. Ginzburg, Representations of quantum tori and double-affine Hecke algebras, preprint, arXiv:math/0005024; an abridged version Representations of quantum tori andG-bundles on elliptic curves published in Progr. Math. 213 (2003), 29-48. · Zbl 1161.17310 |
| [6] | A. Braverman, M. Finkelberg, Finite difference quantum Toda lattice via equivariantK-theory, Transf. Groups 10 (2005), no. 3-4, 363-386. · Zbl 1122.17008 · doi:10.1007/s00031-005-0402-4 |
| [7] | A. Braverman, B. Feigin, M. Finkelberg, L. Rybnikov, A finite analog of the AGT relation I: Finite W-algebras and quasimaps’ spaces, Comm. Math. Phys. 308 (2011), no. 2, 457-478. · Zbl 1247.81169 · doi:10.1007/s00220-011-1300-3 |
| [8] | R. Bezrukavnikov, M. Finkelberg, I. Mirković, Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compos. Math. 141 (2005), no. 3, 746-768. · Zbl 1065.19004 · doi:10.1112/S0010437X04001228 |
| [9] | A. Braverman, M. Finkelberg, H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \({\mathcal{N}}=4\) gauge theories, II, Adv. Theor. Math. Phys. 22 (2018), no. 5, 1071-1147; arXiv:1601.03586. · Zbl 1479.81043 |
| [10] | A. Braverman, M. Finkelberg, H. Nakajima, Coulomb branches of 3d \({\mathcal{N}}=4\) quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, Alex Weekes), preprint, arXiv:1604.03625. |
| [11] | M. Bertola, M. Gekhtman, Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions, Constr. Approx. 26 (2007), no. 3, 383-430. · Zbl 1149.42012 · doi:10.1007/s00365-006-0656-1 |
| [12] | J. Brundan and A. Kleshchev, Shifted Yangians and finiteW-algebras, Adv. Math. 200 (2006), no. 1, 136-195. · Zbl 1083.17006 · doi:10.1016/j.aim.2004.11.004 |
| [13] | I. Cherednik, B. Feigin, Rogers-Ramanujan type identities and Nil-DAHA, Adv. Math. 248 (2013), 1050-1088. · Zbl 1298.33029 · doi:10.1016/j.aim.2013.08.025 |
| [14] | S. Cherkis, A. Kapustin, Periodic monopoles with singularities and \({\mathcal{N}}=2\) super-QCD, Comm. Math. Phys. 234 (2003), no. 1, 1-35. · Zbl 1040.53032 · doi:10.1007/s00220-002-0786-0 |
| [15] | S. Cautis, H. Williams, Cluster theory of the coherent Satake category, preprint, arXiv:1801.08111. · Zbl 1442.22022 |
| [16] | V. Drinfeld, A New realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988), no. 2, 212-216. · Zbl 0667.16004 |
| [17] | J. Ding, I. Frenkel, Isomorphism of two realizations of quantum affine algebra \(U_q(\widehat{\mathfrak{gl}(n)})\), Comm. Math. Phys. 156 (1993), no. 2, 277-300. · Zbl 0786.17008 |
| [18] | P. Di Francesco, R. Kedem, QuantumQ-systems: from cluster algebras to quantum current algebras, Lett. Math. Phys 107 (2017), no. 2, 301-341. · Zbl 1367.81087 · doi:10.1007/s11005-016-0902-2 |
| [19] | P. Etingof, Whittaker functions on quantum groups andq-deformed Toda operators, Amer. Math. Soc. Transl. Ser. 2 194 (1999), 9-25. · Zbl 1157.33327 |
| [20] | S. Evens, J.-H. Lu, Poisson geometry of the Grothendieck resolution of a complex semisimple group, Moscow Math. Journal 7 (2007), no. 4, 613-642. · Zbl 1148.53061 · doi:10.17323/1609-4514-2007-7-4-613-642 |
| [21] | L. Faybusovich, M. Gekhtman, Elementary Toda orbits and integrable lattices, J. Math. Phys. 41 (2000), no. 5, 2905-2921. · Zbl 1052.37051 · doi:10.1063/1.533279 |
| [22] | B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian, Lett. Math. Phys. 88 (2009), no. 1-3, 39-77; correction to “Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian”, Lett. Math. Phys. 108 (2018), no. 7, 1779-1781. · Zbl 1407.37102 |
| [23] | B. Feigin, M. Finkelberg, A. Negut, L. Rybnikov, Yangians and cohomology rings of Laumon spaces, Selecta Math. (N.S.) 17 (2011), no. 3, 573-607. · Zbl 1260.14015 |
| [24] | M. Finkelberg, J. Kamnitzer, K. Pham, L. Rybnikov, A. Weekes, Comultiplication for shifted Yangians and quantum open Toda lattice, Adv. Math. 327 (2018), 349-389. · Zbl 1384.81045 · doi:10.1016/j.aim.2017.06.018 |
| [25] | M. Finkelberg, A. Kuznetsov, L. Rybnikov, Towards a cluster structure on trigonometric zastava (with appendix by Galyna Dobrovolska), Selecta Math. (N.S.) 24 (2018), no. 1, 187-225. · Zbl 1423.13122 |
| [26] | V. Fock, A. Marshakov, A note on quantum groups and relativistic Toda theory, Nuclear Phys. B Proc. Suppl. 56B (1997), 208-214. · Zbl 0957.37516 · doi:10.1016/S0920-5632(97)00328-9 |
| [27] | V. Futorny, A. Molev, S. Ovsienko, Gelfand-Tsetlin bases for representations of finite W-algebras and shifted Yangians, in “Lie theory and its applications in physics VII”, (H. D. Doebner and V. K. Dobrev, Eds), Proceedings of the VII International Workshop, Varna, Bulgaria, June 2007. Heron Press, Sofia (2008), 352-363. · Zbl 1202.81093 |
| [28] | L. Faddeev, L. Takhtajan, The quantum method for the inverse problem and the XYZ Heisenberg model, Russian Math. Surveys 34 (1979), no. 5, 11-68. · Zbl 0449.35096 · doi:10.1070/RM1979v034n05ABEH003909 |
| [29] | M. Finkelberg, A. Tsymbaliuk, Shifted quantum affine algebras: integral forms in typeA (with appendices by Alexander Tsymbaliuk and Alex Weekes), Arnold Mathematical Journal (2019), https://doi.org/10.1007/s40598-019-00118-7; arXiv:1811.12137. · Zbl 1469.17014 · doi:10.1007/s40598-019-00118-7 |
| [30] | A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, On a class of representations of the Yangian and moduli space of monopoles, Comm. Math. Phys. 260 (2005), no. 3, 511-525. · Zbl 1142.53069 · doi:10.1007/s00220-005-1417-3 |
| [31] | A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, On a class of representations of quantum groups, Noncommutative geometry and representation theory in mathematical physics, Contemp. Math. 391, Amer. Math. Soc., Providence, RI (2005), 101-110. · Zbl 1104.17008 |
| [32] | L. Gow, A. Molev, Representations of twistedq-Yangians, Selecta Math. (N.S.) 16 (2010), no. 3, 439-499. · Zbl 1206.81060 |
| [33] | N. Guay, H. Nakajima, C. Wendlandt, Coproduct for Yangians of affine Kac-Moody algebras, Adv. in Math. 338 (2018), 865-911. · Zbl 1447.17012 · doi:10.1016/j.aim.2018.09.013 |
| [34] | M. Gekhtman, M. Shapiro, A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective, Acta Math. 206 (2011), no. 2, 245-310. · Zbl 1228.53095 · doi:10.1007/s11511-011-0063-1 |
| [35] | R. Gonin, A. Tsymbaliuk, On Sevostyanov’s construction of quantum difference Toda lattices, IMRN (2019), https://doi.org/10.1093/imrn/rnz083; arXiv:1804.01063. · Zbl 1491.17011 · doi:10.1093/imrn/rnz083 |
| [36] | M. Haiman, Cherednik algebras, Macdonald polynomials and combinatorics, ICM 2006 Proceedings 3, European Math. Soc. (2006), 843-872. · Zbl 1099.33014 |
| [37] | D. Hernandez, Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), no. 2, 163-200. · Zbl 1102.17009 · doi:10.1007/s00031-005-1005-9 |
| [38] | T. Hoffmann, J. Kellendonk, N. Kutz, N. Reshetikhin, Factorization dynamics and Coxeter-Toda lattices, Comm. Math. Phys. 212 (2000), no. 2, 297-321. · Zbl 0989.37074 · doi:10.1007/s002200000212 |
| [39] | X. He, G. Lusztig, A generalization of Steinberg’s cross section, J. Amer. Math. Soc. 25 (2012), no. 3, 739-757. · Zbl 1252.20047 · doi:10.1090/S0894-0347-2012-00728-0 |
| [40] | M. Jimbo, Aq-analogue of \(U({\mathfrak{gl}}(N+1))\), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247-252. · Zbl 0602.17005 |
| [41] | N. Jing, On Drinfeld realization of quantum affine algebras, in Monster and Lie Algebras, Columbus, OH, 1996 (Ohio State University Mathematical Research Institute Publications, de Gruyter, Berlin, 1998), Vol. 7, 195-206. · Zbl 0983.17013 |
| [42] | A. Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 29, Springer-Verlag, Berlin (1995). · Zbl 0808.17004 · doi:10.1007/978-3-642-78400-2 |
| [43] | V. Kuznetsov, A. Tsyganov, Quantum relativistic Toda chains, Journal of Math. Sciences 80 (1996), no. 3, 1802-1810. · Zbl 0860.17049 · doi:10.1007/BF02362778 |
| [44] | J. Kamnitzer, P. Tingley, B. Webster, A. Weekes, O. Yacobi, Highest weights for truncated shifted Yangians and product monomial crystals, preprint, arXiv:1511.09131. · Zbl 1448.16023 |
| [45] | J. Kamnitzer, B. Webster, A. Weekes, O. Yacobi, Yangians and quantizations of slices in the affine Grassmannian, Algebra Number Theory 8 (2014), no. 4, 857-893. · Zbl 1325.14068 · doi:10.2140/ant.2014.8.857 |
| [46] | G. Laumon, Un analogue global du cône nilpotent, Duke Math. Journal 57 (1988), no. 2, 647-671. · Zbl 0688.14023 · doi:10.1215/S0012-7094-88-05729-8 |
| [47] | S. Levendorskii, On generators and defining relations of Yangians, Journal of Geometry and Physics 12 (1993), no. 1, 1-11. · Zbl 0785.17019 · doi:10.1016/0393-0440(93)90084-R |
| [48] | J.-H. Lu, Hopf algebroids and quantum groupoids, International J. Math. 7 (1996), no. 1, 47-70. · Zbl 0884.17010 · doi:10.1142/S0129167X96000050 |
| [49] | G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976), no. 2, 101-159. · Zbl 0366.20031 · doi:10.1007/BF01408569 |
| [50] | G. Lusztig, Introduction to quantum groups, Progress in Math. 110 (1993). · Zbl 0788.17010 |
| [51] | A. Molev, Yangians and classical Lie algebras, Mathematical Surveys and Monographs, 143, American Mathematical Society, Providence, RI (2007). · Zbl 1141.17001 · doi:10.1090/surv/143 |
| [52] | A. Malkin, V. Ostrik, M. Vybornov, The minimal degeneration singularities in the affine Grassmannians, Duke Math. J. 126 (2005), no. 2, 233-249. · Zbl 1078.14016 · doi:10.1215/S0012-7094-04-12622-3 |
| [53] | A. Negut, Quantum toroidal and shuffle algebras, preprint, arXiv:1302.6202. · Zbl 1427.14023 |
| [54] | M. Nazarov, V. Tarasov, Yangians and Gel’fand-Zetlin bases, Publ. RIMS Kyoto Univ. 30 (1994), no. 3, 459-478. · Zbl 0929.17009 · doi:10.2977/prims/1195165907 |
| [55] | G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118-198. · Zbl 1159.22010 · doi:10.1016/j.aim.2008.04.006 |
| [56] | A. Sevostyanov, Quantum deformation of Whittaker modules and the Toda lattice, Duke Math. J. 105 (2000), no. 2, 211-238. · Zbl 1058.17009 · doi:10.1215/S0012-7094-00-10522-4 |
| [57] | A. Sevostyanov, Conjugacy classes in Weyl groups and q-W algebras, Adv. Math. 228 (2011), no. 3, 1315-1376. · Zbl 1229.17017 · doi:10.1016/j.aim.2011.06.018 |
| [58] | R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49-80. · Zbl 0136.30002 · doi:10.1007/BF02684397 |
| [59] | M. Semenov-Tian-Shansky, Poisson Lie groups, quantum duality principle, and the quantum double, Contemp. Math. 175 (1994), 219-248. · Zbl 0841.17008 · doi:10.1090/conm/175/01845 |
| [60] | M. Semenov-Tian-Shansky, A. Sevostyanov, Drinfeld-Sokolov reduction for difference operators and deformations of \({\mathcal{W}} \)-algebras. II. The general semisimple case, Comm. Math. Phys. 192 (1998), no. 3, 631-647. · Zbl 0916.17021 |
| [61] | A. Tsymbaliuk, Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra viaK-theory of affine Laumon spaces, Selecta Math. (N.S.) 16 (2010), no. 2, 173-200; for a corrected version refer to arXiv:0903.0917v3. · Zbl 1284.17011 |
| [62] | A. Tsymbaliuk, The affine Yangian of \(\mathfrak{gl}_1\) revisited, Adv. Math. 304 (2017), 583-645. · Zbl 1350.81017 · doi:10.1016/j.aim.2016.08.041 |
| [63] | A. Tsymbaliuk, PBWD bases and shuffle algebra realizations for \(U_{\boldsymbol{v}}(L\mathfrak{sl}_n), U_{{\boldsymbol{v}}_1,{\boldsymbol{v}}_2}(L\mathfrak{sl}_n), U_{\boldsymbol{v}}(L\mathfrak{sl}(m|n))\) and their integral forms, preprint, arXiv:1808.09536. |
| [64] | M. Varagnolo, E. Vasserot, Double affine Hecke algebras and affine flag manifolds, I. Affine flag manifolds and principal bundles, Trends in Mathematics, Birkhäuser/Springer, Basel (2010), 233-289. · Zbl 1242.20007 |
| [65] | M. · Zbl 1280.20005 · doi:10.1090/S1088-4165-2010-00384-2 |
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