Representation of elliptic Ding-Iohara algebra

Abstract

We define a vector representation V (u) of elliptic Ding-Iohara algebra U (q; t; p): Furthermore, we construct the tensor products of the vector representations and the Fock modules ℱ(u) by taking the inductive limit of certain subspaces in the finite tensor products of vector representations.

Keywords

Elliptic Ding-Iohara algebra / vector representation / partition

References

[1] Awata H,Feigin B, Hoshino A, Kanai M, Shiraishi J,Yanagida S. Notes on Ding-Iohara algebra and AGT conjecture. arXiv:1106.4088

[2] Awata H, Feigin B, Shiraishi J. Quantum algebraic approach to refined topological vertex. J High Energy Phys, 2012, 03: 041

[3] Awata H, Yamada Y. Five-dimensional AGT conjecture and the Deformed Virasoro algebra. J High Energy Phys, 2010, 01: 125

[4] AwataH, Yamada Y. Five-dimensional AGT relation and the Deformed β-ensemble. Progr Theor Phys, 2010, 124: 227–262

[5] Cai L Q, Wang L F, Wu K, Yang J. Operator product formulas in the algebraic approach of the refined topological vertex. Chin Phys Lett, 2013, 30: 020306

[6] Cai L Q, Wang L F, Wu K, Yang J. Fermion representation of quantum toroidal algebra. Commun Theor Phys (Beijing), 2014, 61: 403–408

[7] Cai L Q, Wang L F, Wu K, Yang J. The Fermion representation of quantum toroidal algebra on 3D Young diagrams. Chin Phys Lett, 2014, 31: 070502

[8] Ding J, Iohara K. Generalization of Drinfeld quantum affne algebras. Lett Math Phys, 1997, 41: 181–193

[9] Feigin B, Feigin E, Jimbo M, Miwa T,Mukhin E. Quantum continuous gl_∞: Semi-infinite construction of representations. Kyoto J Math, 2011, 51: 337–364

[10] Feigin B, Hashizume K, Hoshino A, Shiraishi J, Yanagida S. A commutative algebra on degenerate ℂP¹ and Macdonald polynomials. J Math Phys, 2009, 50: 095215

[11] Feigin B, Jimbo M, Miwa T, Mukhin E. Quantum toroidal gl₁ algebra: plane partitions. Kyoto J Math, 2012, 52: 621–659

[12] Komori Y, Noumi M, Shiraishi J. Kernel functions for difference operators of Ruijsenaars type and their applications. SIGMA Symmetry Integrability Geom Methods Appl, 2009, 5: 054

[13] Langmann E. Second quantization of the elliptic Calogero-Sutherland model. Comm Math Phys, 2004, 247: 321–351

[14] Langmann E. Remarkable identities related to the (quantum) elliptic Calogero-Sutheland model. J Math Phys, 2006, 47: 022101

[15] Miki K. A (q; γ) analog of the W_1+∞ algebra. J Math Phys, 2007, 48: 123520

[16] Nekrasov N, Pestun V. Seiberg-Witten geometry of four dimensional N= 2 quiver gauge theories. arXiv: 1211.2240

[17] Nekrasov N, Pestun V, Shatashvili S. Quantum geometry and quiver gauge theories. arXiv: 1312.6689

[18] Saito Y. Elliptic Ding-Iohara algebra and the free field realization of the Elliptic Macdonald operator. Publ Res Inst Math Sci, 2014, 50(3): 411–455

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