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Quantum toroidal algebra associated with \(\mathfrak{gl}_{m|n}\). (English) Zbl 1486.17023

The quantum toroidal algebra associated with \(\mathfrak{gl}_m\) has been found to have many applications in various branches of mathematics and mathematical physics. In the article under review, the authors introduce the quantum toroidal algebra \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) associated with \(\mathfrak{gl}_{m|n}\) with \(m \neq n\) and \(q_1q_2q_3 = 1\), which generalizes the quantum toroidal algebra \(\mathcal{E}_{m|0}(q_1,q_2,q_3)\) associated with \(\mathfrak{gl}_m\). The authors explain that they expect that the algebra \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) have many properties similar to \(\mathcal{E}_{m|0}(q_1,q_2,q_3)\). In particular, they construct the evaluation map; a surjective algebra homomorphism from \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) to the quantum affine algebra associated with \(\mathfrak{gl}_{m|n}\).

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
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[1] Awata, H.; Feigin, B.; Shiraishi, J., Quantum algebraic approach to refined topological vertex, J. High Energy Phys., 2012, 3, 41-68 (2012) · Zbl 1309.81112 · doi:10.1007/JHEP03(2012)041
[2] Bezerra, L., Mukhin, E: Braid actions on quantum toroidal superalgebras, arXiv:1912.08729
[3] Burban, I.; Schiffmann, O., On the Hall algebra of an elliptic curve I, Duke Math. J., 161, 7, 1171-1231 (2012) · Zbl 1286.16029 · doi:10.1215/00127094-1593263
[4] Ding, J.; Iohara, K., Generalization of drinfeld quantum affine algebras, Lett. Math. Phys., 41, 2, 181-193 (1997) · Zbl 0889.17011 · doi:10.1023/A:1007341410987
[5] Feigin, B.; Jimbo, M.; Mukhin, E., Integrals of motion from quantum toroidal algebras, J. Phys. A: Math. Theor., 50, 464001 (2017) · Zbl 1386.82037 · doi:10.1088/1751-8121/aa8e92
[6] Feigin, B., Jimbo, M., Mukhin, E.: An evaluation homomorphism for quantum toroidal \(\mathfrak{gl}(n)\) algebras, arXiv:1709.01592v2
[7] Feigin, B., Jimbo, M., Mukhin, E.: Towards trigonometric deformation of \(\widehat{\mathfrak{sl}}_2\) coset VOA, arXiv:1811.02056v1 · Zbl 1416.81093
[8] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Representations of quantum toroidal \(\mathfrak{gl}_N\) glN, J. Algebra, 380, 78-108 (2013) · Zbl 1293.17017 · doi:10.1016/j.jalgebra.2012.12.029
[9] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Branching rules for quantum toroidal \(\mathfrak{gl}_N\) glN, Adv. Math., 300, 229-274 (2016) · Zbl 1402.17031 · doi:10.1016/j.aim.2016.03.019
[10] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., Finite type modules and Bethe ansatz for the quantum toroidal \(\mathfrak{gl}_1\) gl1, Ann. Henri Poincaré, 18, 8, 2543-2579 (2017) · Zbl 1407.82019 · doi:10.1007/s00023-017-0577-y
[11] Feigin, B.; Tsymbaliuk, A., Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra, Kyoto J. Math., 51, 4, 831-854 (2011) · Zbl 1242.14006 · doi:10.1215/21562261-1424875
[12] Ginzburg, V.; Kapranov, M.; Vasserot, E., Langlands reciprocity for algebraic surfaces, Math. Res. Lett., 2, 2, 147-160 (1995) · Zbl 0914.11040 · doi:10.4310/MRL.1995.v2.n2.a4
[13] Kojima, T., A bosonization of \(U_q(\widehat{\mathfrak{sl}}_{m|n})\) Uq(sl ̂m|n), Comm. Math. Phys., 355, 2, 603-644 (2017) · Zbl 1431.81078 · doi:10.1007/s00220-017-2957-z
[14] Kojima, T., Commutation relations of vertex operators for \(u_q(\widehat{\mathfrak{sl}}_{m|n})\) uq(sl ̂m|n), J. Math. Phys., 59, 10, 101701,37 (2018) · Zbl 1408.81019 · doi:10.1063/1.5047255
[15] Kimura, K.; Shiraishi, J.; Uchiyama, J., A level-one representation of the quantum affine superalgebra \(U_q(\widehat{\mathfrak{sl}}(M + 1|N + 1))\) Uq(sl ̂(M + 1|N + 1)), Comm. Math. Phys., 188, 2, 367-378 (1997) · Zbl 0899.17004 · doi:10.1007/s002200050169
[16] Kac, VG; Wakimoto, M., Integrable Highest Weight Modules over Affine Superalgebras and Appell’s Function, Comm. Math. Phys., 215, 3, 631-682 (2001) · Zbl 0980.17002 · doi:10.1007/s002200000315
[17] Miki, K., Toroidal braid group action and an automorphism of toroidal algebra \(U_q\bigl (\mathfrak{sl}_{n + 1,tor}\bigr )\) Uq(sln+ 1,tor) (n ≥ 2), Lett. Math. Phys., 47, 4, 365-378 (1999) · Zbl 1022.17009 · doi:10.1023/A:1007556926350
[18] Miki, K., Toroidal and level \(0 u_q^{\prime }\widehat{sl_{n + 1}}\) uq′sln+ 1 ̂ actions on \(u_q\widehat{gl_{n + 1}}\) uqgln+ 1 ̂ modules, J. Math. Phys., 40, 6, 3191-3210 (1999) · Zbl 0959.17016 · doi:10.1063/1.533078
[19] Negut, A., The Shuffle Algebra Revisited, Int. Math. Res. Not., 2014, 22, 6242-6275 (2014) · Zbl 1310.16030 · doi:10.1093/imrn/rnt156
[20] Saito, Y., Quantum toroidal algebras and their vertex representations, Publ. Res. Inst. Math. Sci., 34, 2, 155-177 (1998) · Zbl 0982.17008 · doi:10.2977/prims/1195144759
[21] Schiffmann, O.; Vasserot, E., The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math., 147, 1, 188-234 (2011) · Zbl 1234.20005 · doi:10.1112/S0010437X10004872
[22] Schiffmann, O.; Vasserot, E., The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of \(\mathbb{A}^2\) A2, Duke Math. J., 162, 2, 279-366 (2013) · Zbl 1290.19001 · doi:10.1215/00127094-1961849
[23] Tsymbaliuk, A., Quantum affine Gelfand—Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces, Sel. Math. New Ser., 16, 2, 173-200 (2010) · Zbl 1284.17011 · doi:10.1007/s00029-009-0013-3
[24] Tsymbaliuk, A: PBWD bases and shuffle algebra realizations for \(U_{\boldsymbol{v},}(L\mathfrak{sl}_n), U_{\boldsymbol{v_1},\boldsymbol{v_2}}(L\mathfrak{sl}_n), U_{\boldsymbol{v}}(L\mathfrak{sl}_{m|n})\), arXiv:1808.09536
[25] Varagnolo, M.; Vasserot, E., Schur duality in the toroidal setting, Comm. Math. Phys., 182, 2, 469-483 (1996) · Zbl 0879.17007 · doi:10.1007/BF02517898
[26] Yamane, H., On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. RIMS Kyoto Univ., 35, 321-390 (1999) · Zbl 0987.17007 · doi:10.2977/prims/1195143607
[27] Zhang, Y., Comments on the Drinfeld realization of the quantum affine superalgebra uq[gl(m|n)(1)] and its Hopf algebra structure, J. Phys. A: Math. Gen., 30, 8325-8335 (1997) · Zbl 1041.17502 · doi:10.1088/0305-4470/30/23/028
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