Quasi Modules for the Quantum Affine Vertex Algebra in Type A

Abstract

We consider the quantum affine vertex algebra \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\) associated with the rational R-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras \({\textrm{A}_c (\mathfrak{gl}_N)}\) of the completed double Yangian \({\widetilde{\textrm{DY}}_{c}(\mathfrak{gl}_N)}\) at the level \({c\in\mathbb{C}}\), associated with the reflection equation, and we employ their structure to construct examples of quasi \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\)-modules. Finally, we use the quasi module map, together with the explicit description of the center of \({\mathcal{V}_{c}(\mathfrak{gl}_N)}\), to obtain formulae for families of central elements in the completed algebra \({\widetilde{\textrm{A}}_c (\mathfrak{gl}_N)}\).

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References

  1. 1

    Cherednik I.V.: Factorizing particles on a half line and root systems. Theor. Math. Phys. 61, 977–983 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Chervov, A.V., Molev, A.I.: On higher order Sugawara operators. Int. Math. Res. Not. 1612–1635 (2009)

  3. 3

    Chervov, A., Talalaev, D.: Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence. arXiv:hep-th/0604128

  4. 4

    Etingof P., Kazhdan D.: Quantization of Lie bialgebras V. Sel. Math. (N.S.) 6, 105–130 (2000)

    Article  MATH  Google Scholar 

  5. 5

    Feigin B., Frenkel E.: Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras. Int. J. Mod. Phys. A 7(Suppl. 1A), 197–215 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Frappat L., Jing N., Molev A., Ragoucy E.: Higher Sugawara operators for the quantum affine algebras of type A. Commun. Math. Phys. 345, 631–657 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. 7

    Frenkel, E.: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, 103. Cambridge University Press, Cambridge (2007)

  8. 8

    Iohara K.: Bosonic representations of Yangian double \({DY_{\hbar}(\mathfrak{g})}\) with \({\mathfrak{g}=\mathfrak{g}\mathfrak{l}_N,\mathfrak{s}\mathfrak{l}_N}\). J. Phys. A 29, 4593–4621 (1996)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9

    Jing N., Kožić S., Molev A., Yang F.: Center of the quantum affine vertex algebra in type A. J. Algebra 496, 138–186 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Jucys A.: On the Young operators of the symmetric group. Lietuvos Fizikos Rinkinys 6, 163–180 (1966)

    MathSciNet  Google Scholar 

  11. 11

    Kulish P.P., Sklyanin E.K.: Algebraic structures related to reflection equations. J. Phys. A 25, 5963–5975 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12

    Kuznetsov V.B., Jørgensen M.F., Christiansen P.L.: New boundary conditions for integrable lattices. J. Phys. A 28, 4639–4654 (1995)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Li H.-S.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules. Contemp. Math. 193, 203–236 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Li H.-S.: Modules-at-infinity for quantum vertex algebras. Commun. Math. Phys. 282, 819–864 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 15

    Li H.-S.: \({\hbar}\)-Adic quantum vertex algebras and their modules. Commun. Math. Phys. 296, 475–523 (2010)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Li H.-S., Tan S., Wang Q.: Twisted modules for quantum vertex algebras. J. Pure Appl. Algebra 214, 201–220 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Mintchev M., Ragoucy E., Sorba P.: Spontaneous symmetry breaking in the gl(N)−NLS hierarchy on the half line. J. Phys. A 34, 8345–8364 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Molev, A.: Yangians and classical Lie algebras, Mathematical Surveys and Monographs, 143. American Mathematical Society, Providence, RI (2007)

  19. 19

    Molev A.I., Ragoucy E.: Representations of reflection algebras. Rev. Math. Phys. 14, 317–342 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20

    Noumi M.: Macdonald’s symmetric polynomials as zonal spherical functions on quantum homogeneous spaces. Adv. Math. 123, 16–77 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Okounkov A.: Quantum immanants and higher Capelli identities. Transform. Groups 1, 99–126 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22

    Reshetikhin N.Y., Semenov-Tian-Shansky M.A.: Central extensions of quantum current groups. Lett. Math. Phys. 19, 133–142 (1990)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Sklyanin E.K.: Boundary conditions for integrable quantum systems. J. Phys. A 21, 2375–2389 (1988)

    ADS  MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The author would like to thank Alexander Molev for fruitful discussions. We would also like to thank the anonymous referee for useful comments and suggestions which helped us to improve the manuscript. The research was partially supported by the Australian Research Council and by the Croatian Science Foundation under the Project 2634.

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Correspondence to Slaven Kožić.

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Dedicated to Mirko Primc on the occasion of his 70th birthday

Communicated by Y. Kawahigashi

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Kožić, S. Quasi Modules for the Quantum Affine Vertex Algebra in Type A. Commun. Math. Phys. 365, 1049–1078 (2019). https://doi.org/10.1007/s00220-019-03291-0

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