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Communications in Mathematical Physics

, Volume 367, Issue 2, pp 455–481 | Cite as

The \({(\mathfrak{gl}_{m},\mathfrak{gl}_{n})}\) Duality in the Quantum Toroidal Setting

  • B. Feigin
  • M. Jimbo
  • E. MukhinEmail author
Article
  • 43 Downloads

Abstract

On a Fock space constructed from mn free bosons and lattice \({\mathbb{Z}_{mn}}\), we give a level n action of the quantum toroidal algebra \({\mathcal{E}_{m}}\) associated to \({\mathfrak{gl}_{m}}\), together with a level m action of the quantum toroidal algebra \({\mathcal{E}_n}\) associated to \({\mathfrak{gl}_n}\). We prove that the \({\mathcal{E}_m}\) transfer matrices commute with the \({\mathcal{E}_n}\) transfer matrices after an appropriate identification of parameters.

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Notes

Acknowledgements

The research of BF is supported by the Russian Science Foundation grant project 16-11-10316. MJ is partially supported by JSPS KAKENHI Grant Number JP16K05183. EM is partially supported by a grant from the Simons Foundation #353831.

EM and BF would like to thank Kyoto University for hospitality during their visits when this work was started.

References

  1. BLZ.
    Bazhanov V., Lukyanov S., Zamolodchikov A.: Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz. Commun. Math. Phys. 177(2), 381–398 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. BLZ1.
    Bazhanov V., Lukyanov S., Zamolodchikov A.: Integrable structure of conformal field theory II. Q-operators and DDV equation. Commun. Math. Phys. 190(2), 247–278 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. BLZ2.
    Bazhanov V., Lukyanov S., Zamolodchikov A.: Integrable structure of conformal field theory III. The Yang-Baxter relation. Commun. Math. Phys. 200(2), 297–324 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. FFR.
    Feigin B., Frenkel E., Reshetikhin N.: Gaudin model, Bethe ansatz and critical level. Commun. Math. Phys. 166(1), 27–62 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. FJM.
    Feigin B., Jimbo M., Mukhin E.: Integrals of motion from quantum toroidal algebras. J. Phys. A Math. Theor. 50, 464001 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. FJMM.
    Feigin B., Jimbo M., Miwa T., Mukhin E.: Branching rules for quantum toroidal \({\mathfrak{gl}_N}\). Adv. Math. 300, 229–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. FJMM1.
    Feigin B., Jimbo M., Miwa T., Mukhin E.: Quantum toroidal \({\mathfrak{gl}_1}\) and Bethe ansatz. J. Phys. A Math. Theor. 48, 244001 (2015)ADSCrossRefzbMATHGoogle Scholar
  8. FJMM2.
    Feigin B., Jimbo M., Miwa T., Mukhin E.: Finite type modules and Bethe ansatz for the quantum toroidal \({\mathfrak{gl}_1}\). Commun. Math. Phys. 356(1), 285–327 (2017)ADSCrossRefzbMATHGoogle Scholar
  9. FJMM3.
    Feigin B., Jimbo M., Miwa T., Mukhin E.: Representations of quantum toroidal \({\mathfrak{gl}_n}\). J. Algebra 380, 78–108 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. FKSW.
    Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The integrals of motion for the deformed Virasoro algebra. arXiv:0705.0427v2
  11. FKSW1.
    Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The integrals of motion for the deformed W algebra \({W_{q,t}\big(\widehat{\mathfrak{sl}}_N\big)}\). arXiv:0705.0627v1
  12. KS.
    Kojima T., Shiraishi J.: The integrals of motion for the deformed W algebra \({W_{q,t} \big(\widehat{\mathfrak{sl}}_N\big)}\) II: Proof of the commutation relations. Commun. Math. Phys. 283(3), 795–851 (2008)ADSCrossRefzbMATHGoogle Scholar
  13. Mi.
    Miki K.: Toroidal braid group action and an automorphism of toroidal algebra \({U_q\big(\mathfrak{sl}_{n+1,tor} \big) (n \geq 2)}\). Lett. Math. Phys. 47(4), 365–378 (1999)MathSciNetCrossRefGoogle Scholar
  14. MTV.
    Mukhin, E., Tarasov, V., Varchenko, A. (2009) A Generalization of the Capelli identity, Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin. Vol. II, Progr. Math., 270, Birkhauser Boston, Inc., Boston, MA, 383–398Google Scholar
  15. MTV1.
    Mukhin E., Tarasov V., Varchenko A.: Bispectral and \({(\mathfrak{gl}_{N}, \mathfrak{gl}_M)}\) dualities. Funct. Anal. Other Math. 1(1), 47–69 (2006)MathSciNetCrossRefGoogle Scholar
  16. MTV2.
    Mukhin E., Tarasov V., Varchenko A.: Bispectral and \({(\mathfrak{gl}_{N}, \mathfrak{gl}_{M})}\) dualities, discrete versus differential. Adv. Math. 218(1), 215–265 (2008)CrossRefGoogle Scholar
  17. Sa.
    Saito Y.: Quantum toroidal algebras and their vertex representations. Publ. RIMS Kyoto Univ. 34(2), 155–177 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. STU.
    Saito Y., Takemura K., Uglov D.: Toroidal actions on level 1 modules for \({U_q(\hat{\mathfrak{sl}}_n)}\). Transform. Groups 3(1), 75–102 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Research University Higher School of Economics, Russian Federation, International Laboratory of Representation Theory and Mathematical PhysicsMoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  3. 3.Department of MathematicsRikkyo UniversityTokyoJapan
  4. 4.Department of MathematicsIndiana University-Purdue University-IndianapolisIndianapolisUSA

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