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Center of the Yangian double in type A

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Abstract

We prove that the R-matrix and Drinfeld presentations of the Yangian double in type A are isomorphic. The central elements of the completed Yangian double in type A at the critical level are constructed. The images of these elements under a Harish-Chandra-type homomorphism are calculated by applying a version of the Poincaré-Birkhoff-Witt theorem for the R-matrix presentation. These images coincide with the eigenvalues of the central elements in the Wakimoto modules.

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References

  1. Bernard D, LeClair A. The quantum double in integrable quantum field theory. Nuclear Phys B, 1993, 399: 709–748

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  3. Crampé N. Hopf structure of the Yangian Y(sln) in the Drinfeld realization. J Math Phys, 2004, 45: 434–447

    Article  ADS  MathSciNet  Google Scholar 

  4. Ding J, Etingof P. The center of a quantum affine algebra at the critical level. Math Res Lett, 1994, 1: 469–480

    Article  MathSciNet  Google Scholar 

  5. Ding J, Frenkel I B. Isomorphism of two realizations of quantum affine algebra \({U_q}(\widehat {gl}(n))\). Comm Math Phys, 1993, 156: 277–300

    Article  ADS  MathSciNet  Google Scholar 

  6. Drinfeld V G. Quantum groups. In: Proceedings of the International Congress of Mathematicians. Providence: Amer Math Soc, 1987, 798–820

    Google Scholar 

  7. Drinfeld V G. A new realization of Yangians and quantized affine algebras. Soviet Math Dokl, 1988, 36: 212–216

    MathSciNet  Google Scholar 

  8. Faddeev L, Reshetikhin N, Takhtajan L. Quantization of Lie groups and Lie algebras. Leningrad Math J, 1990, 1: 193–225

    MathSciNet  Google Scholar 

  9. Faddeev L, Sklyanin E, Takhtajan L. The quantum inverse problem method. I. Theoret Math Phys, 1990, 40: 194–220

    MathSciNet  Google Scholar 

  10. Faddeev L, Takhtajan L. The quantum inverse scattering method of the inverse problem and the Heisenburg XYZ model. Russian Math Surveys, 1979, 34: 11–68

    Article  ADS  Google Scholar 

  11. Frappat L, Jing N, Molev A, et al. Higher Sugawara operators for the quantum affine algebras of type A. Comm Math Phys, 2016, 345: 631–657

    Article  ADS  MathSciNet  Google Scholar 

  12. Gel’fand I M, Retakh V S. Determinants of matrices over noncommutative rings. Funct Anal Appl, 1991, 25: 91–102

    Article  MathSciNet  Google Scholar 

  13. Hou B-Y, Zhao L, Ding X-M. Another free Boson representation of Yangian double DYh(slv) with arbitrary level. Commun Theor Phys, 2001, 35: 167–172

    Article  ADS  Google Scholar 

  14. Iohara K. Bosonic representations of Yangian double \({\rm{D}}{{\rm{Y}}_h}(\mathfrak{g})\) with \(\mathfrak{g} = {\mathfrak{g}\mathfrak{l}_N},{\mathfrak{s}\mathfrak{l}_N}\). J Phys A Math Gen, 1996, 29: 4593–4621

    Article  ADS  Google Scholar 

  15. Jimbo M. A q-difference analogue of \(U(\mathfrak{g})\) and the Yang-Baxter equation. Lett Math Phys, 1985, 10: 63–69

    Article  ADS  MathSciNet  Google Scholar 

  16. Jing N, Kožić S, Molev A, et al. Center of the quantum affine vertex algebra in type A. J Algebra, 2018, 496: 138–186

    Article  MathSciNet  Google Scholar 

  17. Jing N, Yang F, Liu M. Yangian doubles of classical types and their vertex representations. J Math Phys, 2020, 61: 051704

    Article  ADS  MathSciNet  Google Scholar 

  18. Khoroshkin S. Central extension of the Yangian double. In: Algebre Non Commutative, Groupes Quantiques et Invariants.. Séminaires et Congrès, vol. 2. Paris: Soc Math France, 1997, 119–135

    Google Scholar 

  19. Leclair A, Smirnov F A. Infinite quantum group symmetry of fields in massive 2D quantum field theory. Internat J Modern Phys A, 1992, 07: 2997–3022

    Article  ADS  Google Scholar 

  20. Molev A. Factorial supersymmetric Schur functions and super Capelli identities. In: Kirillov’s Seminar on Representation Theory. Advances in the Mathematical Sciences, vol. 35. Providence: Amer Math Soc, 1998, 109–137

    Google Scholar 

  21. Molev A. Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. Providence: Amer Math Soc, 2007

    Google Scholar 

  22. Molev A. Feigin-Frenkel center in types B, C and D. Invent Math, 2013, 191: 1–34

    Article  ADS  MathSciNet  Google Scholar 

  23. Molev A, Mukhin E. Yangian characters and classical W-algebras. In: Kohnen W, Weissauer R, eds. Conformal Field Theory, Automorphic Forms and Related Topics. New York: Springer, 2014, 287–334

    Chapter  Google Scholar 

  24. Smirnov F A. Dynamical symmetries of massive integrable models 1: Form factor bootstrap equations as a special case of deformed Knizhnik-Zamolodchikov equations. Internat J Modern Phys A, 1992, 7: 813–837

    Article  ADS  MathSciNet  Google Scholar 

  25. Smirnov F A. Dynamical symmetries of massive integrable models 2: Space of states of massive models as space of operators. Internat J Modern Phys A, 1992, 7: 839–858

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12101261 and 12171303) and the Simons Foundation (Grant No. 523868).

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Authors and Affiliations

  1. Department of Mathematics, Jiaying University, Meizhou, 514000, China

    Fan Yang

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Naihuan Jing

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  1. Fan Yang
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  2. Naihuan Jing
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Correspondence to Naihuan Jing.

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Yang, F., Jing, N. Center of the Yangian double in type A. Sci. China Math. (2024). https://doi.org/10.1007/s11425-022-2142-9

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