Kyoto Journal of Mathematics

Quantum continuous gl: Semiinfinite construction of representations

B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin

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We begin a study of the representation theory of quantum continuous gl, which we denote by E. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of E are labeled by a continuous parameter uC. The representation theory of E has many properties familiar from the representation theory of gl: vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from E to spherical double affine Hecke algebras SN for all N. A key step in this construction is an identification of a natural basis of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the K-theory of Hilbert schemes.

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Kyoto J. Math., Volume 51, Number 2 (2011), 337-364.

First available in Project Euclid: 22 April 2011

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Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70] 05E10: Combinatorial aspects of representation theory [See also 20C30]


Feigin, B.; Feigin, E.; Jimbo, M.; Miwa, T.; Mukhin, E. Quantum continuous $\mathfrak{gl}_{\infty}$ : Semiinfinite construction of representations. Kyoto J. Math. 51 (2011), no. 2, 337--364. doi:10.1215/21562261-1214375.

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  • [BS] I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve, I, preprint, arXiv:0505148v2 [math.AG]
  • [C] I. Cherednik, Double Affine Hecke Algebras, London Math. Soc. Lecture Note Ser. 319, Cambridge Univ. Press, Cambridge, 2004.
  • [DI] J. Ding and K. Iohara, Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997), 181–193.
  • [FJM1] B. L. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Symmetric polynomials vanishing on the diagonals shifted by roots of unity, Int. Math. Res. Not. 2003, 999–1014.
  • [FJM2] B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials, Int. Math. Res. Not. 2003, 1015–1034.
  • [FO] B. L. Feigin and A. V. Odesskii, “Vector bundles on an elliptic curve and Sklyanin algebras” in Topics in Quantum Groups and Finite-Type Invariants, Amer. Math. Soc., Transl. Ser. (2), 185, Amer. Math. Soc., Providence, 1998, 65–84.
  • [FHH+] B. L. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate $\mathbb{CP}^{1}$ and Macdonald polynomials, J. Math. Phys. 50, (2009), no. 095215.
  • [FT] B. L. Feigin and A. Tsymbaliuk, Heisenberg action in the equivariant K-theory of Hilbert schemes via shuffle algebra, preprint, arXiv:0904.1679v1 [math.RT]
  • [Kap] M. Kapranov, “Eisenstein series and quantum affine algebras” in Algebraic Geometry, 7, J. Math. Sci. (New York) 84, Consultants Bureau, New York, 1311–1360.
  • [Kas] M. Kasatani, Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GLn at tk+1qr−1 = 1, Int. Math. Res. Not. 2005, no. 28, 1717–1742.
  • [M] I. Macdonald, Symmetric Functions and Hall Polynomials, with contributions by A. Zelevinsky, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  • [S] O. Schiffmann, On the Hall algebra of an elliptic curve, II, preprint, arXiv:0508553v2 [math.RT]
  • [SV1] O. Schiffmann and E. Vasserot, The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, preprint, arXiv:0802.4001v1 [math.QA]
  • [SV2] O. Schiffmann, E. Vasserot, The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^{2}$, preprint, arXiv:0905.2555v2 [math.QA]