Algebras and Representation Theory

, Volume 18, Issue 1, pp 103–135 | Cite as

Elliptic Algebra \(U_{q,p}(\widehat {\mathfrak {g}})\) and Quantum Z-algebras

  • Rasha M. Farghly
  • Hitoshi KonnoEmail author
  • Kazuyuki Oshima


A new definition of the elliptic algebra \(U_{q,p}(\widehat {\mathfrak {g}})\) associated with an untwisted affine Lie algebra \(\widehat {\mathfrak {g}}\) is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson’s Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional \(U_{q,p}({\widehat {\mathfrak {g}}})\)-modules. Some level-1 examples indicate a direct connection of the irreducible \(U_{q,p}(\widehat {\mathfrak {g}})\)-modules to those of the W-algebras associated with the coset \(\widehat {\mathfrak {g}}\oplus \widehat {\mathfrak {g}}\supset (\widehat {\mathfrak {g}})_{{diag}}\) with level (rg − 1, 1) (g:the dual Coxeter number), which includes Fateev-Lukyanov’s W B l -algebra.


Quantum group Affine Lie algebra Virasoro algebra W-algebra Z-algebra 

Mathematics Subject Classification (2010)

17B37 20G42 81R10 81R50 


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  1. 1.
    Konno, H.: An elliptic algebra \(U_{q,p}(\widehat {\mathfrak {sl}}_{2})\) and the fusion RSOS models. Comm. Math. Phys. 195, 373–403 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Jimbo, M., Konno, H., Odake, S., Shiraishi, J.: Elliptic algebra \(U_{q,p}(\widehat {\mathfrak {sl}}_{2})\): drinfeld currents and vertex operators. Comm. Math. Phys. 199, 605–647 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988)MathSciNetGoogle Scholar
  4. 4.
    Frønsdal, C.: Quasi-Hopf deformations of quantum groups. Lett. Math. Phys. 40, 117–134 (1997)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Jimbo, M., Konno, H., Odake, S., Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups. Transform. Groups 4, 303–327 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Enriquez, B., Felder, G.: Elliptic quantum groups \(E_{\tau ,\eta }(\mathfrak {sl}_{2})\) and quasi-Hopf algebra. Comm. Math. Phys. 195, 651–689 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kojima, T., Konno, H.: The elliptic algebra \(U_{q,p}(\widehat {\mathfrak {sl}}_{N})\) and the Drinfeld realization of the elliptic quantum group \({\mathcal {B}_{q,\lambda }}(\widehat {\mathfrak {sl}}_{N})\). Comm. Math. Phys. 239, 405–447 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Etingof, P., Varchenko, A.: Solutions of the Quantum dynamical Yang-Baxter equation and dynamical quantum groups. Comm. Math. Phys. 196, 591–640 (1998). Exchange dynamical quantum groups, Comm. Math. Phys. 205, 19–52 1999CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Koelink, E., Rosengren, H.: Harmonic analysis on the S U(2) dynamical quantum group. Acta. Appl. Math. 69, 163–220 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Konno, H.: Elliptic quantum group \(U_{q,p}(\widehat {\mathfrak {sl}}_{2})\), Hopf algebroid structure and elliptic hypergoemetric series. J. Geom. Phys. 59, 1485–1511 (2008)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and Super-Virasoro algebras. Comm. Math. Phys. 103, 105–119 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ravanini, F.: An infinite calss of new conformal field theories with extended algebras. Mod. Phys. Lett. A3, 397–412 (1988)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Kastor, D., Martinec, E., Qiu, Z.: Current algebra and conformal discrete series. Phys. Lett. B 200, 434–440 (1988)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Bagger, J., Nermeschansky, D., Yankielowicz, S.: Virasoro algebras with central charge c > 1. Phys. Rev. Lett 60, 389–392 (1988)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Di Francesco, P., Saleur, H., Zuber, J-B.: Generalized coulomb-gas formalizm for two dimensional critical models based on S U(2) coset construction. Nucl. Phys B300, 393–432 (1988)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gerasimov, A., Marshakov, A., Morozov, A.: Free field representation of parafermions and related coset models. Nucl. Phys. B 328, 664–676 (1989)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Zamolodchikov, A.B., Fateev, V.A.: Representations of the algebra of “parafermion currents” of spin 4/3 in two-dimensional conformal field theory. Minimal models and the tricritical potts \(\mathbb {Z}_{3}\) model. Theor. Math. Phys. 71, 451–462 (1987)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lepowsky, J., Wilson, R.L.: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations. Proc. Natl. Acad. Sci. USA 78, 7254–7258 (1981). The structure of standard modules, I: universal algebras and the Roger-Ramanujan identities, Invent. Math. 77, 199–290 (1984) Google Scholar
  19. 19.
    Lukyanov, S., Pugai, Y.: Multi-point local height probabilities in the integrable RSOS model. Nucl. Phys. B473, 631–658 (1996)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Gepner, D.: New conformal field theories associated with Lie algebras and their partition functions. Nucl. Phys. B290, 10–24 (1987)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Matsuo, A.: A q−deformation of Wakimoto modules, primary fields and screening operators. Comm. Math. Phys. 160, 33–48 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Bougourzi, A., Vinet, L.: A Quantum analog of the \(\mathcal {Z}\) algebra. J. Math. Phys. 37, 3548–3567 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Jing, N.: Higher level representations of the quantum affine algebra \(U_{q}(\widehat {sl}(2))\). J. Algebra 182, 448–468 (1996). Quantum z-algebras and representations of quantum affine algebras, Comm. Alg., 28, 829–844 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jimbo, M., Konno, H., Odake, S., Pugai, Y., Shiraishi, J.: Free field construction for the ABF models in Regime II. J. Stat. Phys. 102, 883–921 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Christe, P., Ravanini, F.: G NG L/G N+L conformal field theories and their modular invariant partition functions. Int. J. Mod. Phys. A4, 897–920 (1989)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Feigin, B., Frenkel, E.: Quantum W-algebras and elliptic algebras. Comm. Math. Phys. 178, 653–678 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum W N algebras and Macdonald polynomials. Comm. Math. Phys. 179, 401–416 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Frenkel, E., Reshetikhin, N.: Deformation of W-algebras associated to simple Lie algebras. q-alg/9708006Google Scholar
  29. 29.
    Kojima, T., Konno, H.: The elliptic algebra \(U_{q,p}(\widehat {\mathfrak {sl}}_{2})\) and the deformation of W N algebra. J. Phys. A37, 371–383 (2004)MathSciNetGoogle Scholar
  30. 30.
    Bouwknegt, P., Schoutens, K.: W symmetry in conformal field theory. Phys. Rep. 223, 183–276 (1993)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Lukyanov, S.L., Fateev, V.A.: Additional symmetries and exactly-soluble models in two-dimensional conformal field theory. Sov. Sci. Rev. A. Phys. 15, 1–117 (1990)Google Scholar
  32. 32.
    Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models. Nucl. Phys. B290, 231–273 (1987)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Jimbo, M., Miwa, T., Okado, M.: Solvable lattice models related to the vector representation of classical simple Lie algebras. Comm. Math. Phys. 116, 507–525 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice models. Conference Board of the Math. Sci. Regional Conference Series in Mathematics 85, 1 (1995). and references thereinGoogle Scholar
  35. 35.
    Kojima, T., Konno, H., Weston, R.: The Vertex-Face Ccorrespondence and correlation functions of the fusion eight-vertex models I: the general formalism. Nucl. Phys. B720, 348–398 (2005)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Konno, H.: Elliptic Quantum Group \(U_{q,p}(\widehat {\mathfrak {sl}}_{2})\) and vertex operators. J. Phys. A 41, 194012 (2008)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Bourbaki, N.: Chaps. 4–6: Groupes et Algebres de Lie. Hermann, Paris (1968)Google Scholar
  38. 38.
    Kac, V.G., 3rd edn.: Infinite Dimensional Lie algebras. Cambridge University Press (1990)Google Scholar
  39. 39.
    Frenkel, I.B., Jing, N.: Vertex representations of Quantum affine algebras. Proc. Nat. Acad. Sci. USA 85, 9373–9377 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Konno, H.: Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations. SIGMA 2, Paper 091 (2006)MathSciNetGoogle Scholar
  41. 41.
    Lepowsky, J., Primc, M.: Standard modules for type one affine Lie algebras. Lec. Note in Math 1052, 194–251 (1984)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Bernard, D.: Vertex operator representations of the quantum affine algebra \(U_{q}(B_{r}^{(1)})\). Lett. Math. Phys. 17, 239–245 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Jing, N., Misra, K.C.: Vertex operators of level-one \(U_{q}(B_{r}^{(1)})\)-modules. Lett. Math. Phys. 36, 127–143 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Idzumi, M.: Level 2 irreducible representations of \(U_{q}(\widehat {\mathfrak {sl}}_{2})\), vertex operators, and their correlations. Int. J. Mod. Phys. A9, 4449–4484 (1994)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Rasha M. Farghly
    • 1
  • Hitoshi Konno
    • 2
    Email author
  • Kazuyuki Oshima
    • 3
  1. 1.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of MathematicsTokyo University of Marine Science and TechnologyKotoJapan
  3. 3.Department of Mathematics, Center for General EducationAichi Institute of TechnologyYakusa-choJapan

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