Skip to main content
SpringerLink
Log in

Deformed Virasoro Algebras from Elliptic Quantum Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 1990s. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the gl(2) case) a Liouville formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Awata H., Kubo H., Odake S., Shiraishi J.: Quantum \({\mathcal{W}_N}\) algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401 (1996) arXiv:hep-th/9508011

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Shiraishi J., Kubo H., Awata H., Odake S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33 (1996) arXiv:q-alg/9507034

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Lukyanov S., Pugai Y.: Multipoint local height probabilities in the integrable RSOS model. Nucl. Phys. B 473, 631 (1996) arXiv:hep-th/9602074

    Article  ADS  MATH  Google Scholar 

  4. Lukyanov S.: A note on the deformed Virasoro algebra. Phys. Lett. B 367, 121 (1996) arXiv:hep-th/9509037

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional Gauge theories. Lett. Math. Phys. 91, 167 (2010) arXiv:0906.3219

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Awata H., Yamada Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 1001, 125 (2010) arXiv:0910.4431

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Nieri, F.: An elliptic Virasoro symmetry in 6d (2015). arXiv:1511.00574

  8. Brazhnikov V., Lukyanov S.: Angular quantization and form-factors in massive integrable models. Nucl. Phys. B 512, 616 (1998) arXiv:hep-th/9707091

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Curtright T.L., Zachos C.K.: Deforming maps for quantum algebras. Phys. Lett. B 243, 237 (1990)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Frenkel E., Reshetikhin N.Y.: Quantum affine algebras and deformations of Virasoro and W algebras. Commun. Math. Phys. 178, 237 (1996) arXiv:q-alg/9505025

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Daskaloyannis C.: Generalized deformed Virasoro algebras. Mod. Phys. Lett. A 07, 809 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Feigin B., Frenkel E.: Quantum W algebras and elliptic algebras. Commun. Math. Phys. 178, 653 (1996) arXiv:q-alg/9508009

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Macdonald, I.G.: Orthogonal polynomials associated with root systems. Sémin. Lothar. Comb. 45, Art. B45a (2000). arXiv:math.QA/0011046

  15. Ruijsenaars S.N.M., Schneider H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (NY) 170, 370 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Avan J., Frappat L., Rossi M., Sorba P.: New \({\mathcal{W}_{q,p}(sl(2))}\) algebras from the elliptic algebra \({\mathcal{ A}_{q,p}(\widehat{sl}(2)_{c})}\). Phys. Lett. A 239, 27 (1998) arXiv:q-alg/9706013

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Avan J., Frappat L., Rossi M., Sorba P.: Deformed \({\mathcal{W}_{N}}\) algebras from elliptic \({sl(N)}\) algebras. Commun. Math. Phys. 199, 697 (1999) arXiv:math.QA/9801105

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Avan J., Frappat L., Rossi M., Sorba P.: Poisson structures on the center of the elliptic algebra \({\mathcal{W}_{q,p}(\widehat{sl}(2)_{c})}\). Phys. Lett. A 235, 323 (1997) arXiv:q-alg/9705012

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Foda O., Iohara K., Jimbo M., Kedem R., Miwa T., Yan H.: An elliptic quantum algebra for \({\widehat{sl}_2}\). Lett. Math. Phys. 32, 259 (1994) arXiv:hep-th/9403094

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Frønsdal, C.: Generalization and exact deformations of quantum groups. Publ. RIMS Kyoto Univ. 33, 91 (1997)

  21. Frønsdal C.: Quasi-Hopf deformations of quantum groups. Lett. Math. Phys. 40, 117 (1997) arXiv:q-alg/9611028

    Article  MathSciNet  MATH  Google Scholar 

  22. Jimbo M., Konno H., Odake S., Shiraishi J.: Quasi-Hopf twistors for elliptic quantum groups. Transform. Groups 4, 303 (1999) arXiv:q-alg/9712029

    Article  MathSciNet  MATH  Google Scholar 

  23. Arnaudon D., Buffenoir E., Ragoucy E., Roche P.: Universal solutions of quantum dynamical Yang–Baxter equations. Lett. Math. Phys. 44, 201 (1998) arXiv:q-alg/9712037

    Article  MathSciNet  MATH  Google Scholar 

  24. Shiraishi J.: Free field constructions for the elliptic algebra \({\mathcal{A}_{q,p}(\widehat{sl}_2)}\) and Baxter’s eight-vertex model. Int. J. Mod. Phys. A 1952, 363 (2004) arXiv:math/0302097

    Article  MathSciNet  MATH  Google Scholar 

  25. Awata, H., Kubo, H., Odake, S., Shiraishi J.: Virasoro-Type Symmetries in Solvable Models, CRM Series in Mathematical Physics. Springer (1996). arXiv:hep-th/9612233

  26. Foda O., Iohara K., Jimbo M., Kedem R., Miwa T., Yan H.: Notes on highest weight modules of the elliptic algebra \({\mathcal{ A}_{q,p}(\widehat{sl}(2)_{c})}\). Prog. Theor. Phys. Suppl. 118, 1 (1995) arXiv:hep-th/9405058

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Belavin A.A.: Dynamical symmetry of integrable quantum systems. Nucl. Phys. B 180, 189 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Chudnovsky D.V., Chudnovsky G.V.: Completely X-symmetric S-matrices corresponding to theta functions. Phys. Lett. A 81, 105 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  29. Tracy C.A.: Embedded elliptic curves and the Yang–Baxter equation. Physica D 16, 203 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Richey M.P., Tracy C.A.: \({\mathbb{Z}_{N}}\) Baxter model: symmetries and the Belavin parametrization. J. Stat. Phys. 42, 311 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  31. Molev, A.: Yangians and Classical Lie Algebras, Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence, RI (2007)

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique et Modélisation (CNRS UMR 8089), Université de Cergy-Pontoise, 95302, Cergy-Pontoise, France

    J. Avan

  2. Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTh, CNRS - Université Savoie Mont Blanc, BP 110, 74941, Annecy-le-Vieux Cedex, France

    L. Frappat & E. Ragoucy

Authors
  1. J. Avan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Frappat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Ragoucy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Frappat.

Additional information

Communicated by H.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avan, J., Frappat, L. & Ragoucy, E. Deformed Virasoro Algebras from Elliptic Quantum Algebras. Commun. Math. Phys. 354, 753–773 (2017). https://doi.org/10.1007/s00220-017-2909-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2909-7

Search

Navigation