Communications in Mathematical Physics

, Volume 283, Issue 3, pp 795–851 | Cite as

The Integrals of Motion for the Deformed W-Algebra \({W_{q,t}(\widehat{gl_N})}\). II. Proof of the Commutation Relations

  • Takeo KojimaEmail author
  • Jun’ichi Shiraishi


We explicitly construct two classes of infinitely many commutative operators in terms of the deformed W-algebra \({W_{q,t}(\widehat{gl_N})}\), and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W 3 algebra [1,2].


Commutation Relation Analytic Continuation Integral Contour Weak Sense Monodromy Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bazhanov V., Lukyanov S., Zamolodchikov Al.: Integral Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz. Commun. Math. Phys. 177(2), No.2, 381–398 (1996)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Bazhanov V., Hibberd A., Khoroshkin S.: Integrable Structure of W 3 Conformal Field Theory. Nucl. Phys. B 622, 475–547 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The Integrals of Motion for the Deformed Virasoro Algebra., 2007
  4. 4.
    Feigin, B., Kojima, T., Shiraishi, J., Watanabe, H.: The Integrals of Motion for the Deformed W-Algebra \({W_{q,t}(\widehat{sl_N})}\). Proceedings for Representation Theory 2006, Atami, Japan, p. 102–114 (2006) [ISBN4-9902328-2-8], available at, 2007
  5. 5.
    Feigin, B., Frenkel, E.: Integrals of Motion and Quantum Groups, Lecture Notes in Mathematics 1620 Integral Systems and Quantum Groups, Berlin: Springer, 1995Google Scholar
  6. 6.
    Shiraishi J., Kubo H., Awata H., Odake S.: A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions. Lett. Math. Phys. 38, 647–666 (1996)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Awata H., Kubo H., Odake S., Shiraishi J.: Quantum W N Algebras and Macdonald Polynomials. Commun. Math. Phys. 179, 401–416 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Feigin B., Frenkel E.: Quantum W-Algebra and Elliptic Algebras. Commun. Math. Phys. 178, 653–678 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Odake, S.: Comments on the Deformed W N Algebra. In APCTP-Nankai Joint Symposium on “Lattice Statistics and Mathematical Physics 2001”, Tianjin China, River Edge, NJ: World Scientific, 2002Google Scholar
  10. 10.
    Feigin B., Odesskii A.: A Family of Elliptic Algebras. Internat. Math. Res. Notices 11, 531–539 (1997)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Asai Y., Jimbo M., Miwa T., Pugai Ya.: Bosoniztion of Vertex Operators for \({A_{n-1}^{(1)}}\) face model. J. Phys. A Math. Gen. 29, 6595–6616 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Feigin B., Jimbo M., Miwa T., Odesskii A., Pugai Ya.: Algebra of Screening Operators for the Deformed W n Algebra. Commun. Math. Phys. 191, 501–541 (1998)zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Kojima T., Konno H.: The Ellptic Algebra \({U_{q,p}(\widehat{sl_N})}\) and the Drinfeld Realization of the Elliptic Quantum Group \({{\mathcal {B}}_{q,\lambda}(\widehat{sl_N})}\). Commun. Math. Phys. 239, 405–447 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Frenkel E., Reshetikhin N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. Commun. Math. Phys. 178, 237–264 (1996)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Frenkel E.: Deformations of the KdV hierarchy and related soliton equations. Internat. Math. Res. Notices 2, 55–76 (1996)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics, College of Science and TechnologyNihon UniversityTokyoJapan
  2. 2.Graduate School of Mathematical ScienceUniversity of TokyoTokyoJapan

Personalised recommendations