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Liouville theory on the lattice and universal exchange algebra for bloch waves

II. Quantum Groups, Symmetries Of Dynamical Systems And Conformal Field Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1510)

Abstract

We review some aspects of Liouville theory and the relation between its integrable and conformal structures. We emphasis its lattice version which exhibits the role of quantum groups.

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References

  1. A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys.Lett. 103B (1981), 207.

    CrossRef  MathSciNet  Google Scholar 

  2. P. Zograf, L. Takhtajan, Action of Liouville equation is a generating function for the accessory parameters and the potential of the Weil-Petersson metric on the Teichmuller space, Func. Anal. Appl. 19 (1985), 219; Math. USSR Sbornik 60 (1988), 143, 297.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J.L. Gervais, A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nucl. Phys. B238 (1984), 125.

    CrossRef  MathSciNet  Google Scholar 

  4. L.D. Faddeev, L. Takhtajan, Liouville model on the lattice, Springer Lecture Notes in Physics 246 (1986), 166.

    CrossRef  MathSciNet  Google Scholar 

  5. O. Babelon, Extended conformal algebra and the Yang-Baxter equation, Phys. Lett. B215 (1988), 523.

    CrossRef  MathSciNet  Google Scholar 

  6. J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B269 (1986), 54.

    CrossRef  MathSciNet  Google Scholar 

  7. A.Alekseev, L.Faddeev, M.Semenov-Tian-Shansky, A.Volkov, The unravelling of the quantum group structure in the WZNW theory, Preprint CERN Th.5981/91.

    Google Scholar 

  8. F.Nijhoff, V.Papageorgiou, H.Capel, Integrable Time-discrete systems: Lattices and Mappings, Preprint Clarkson University INS 166 (1990); Integrable Quantum Mappings, Preprint Clarkson University INS 168 (1991).

    Google Scholar 

  9. A.N. Leznov, M.V. Saveliev, Representation of zero curvature for the system of non-linear partial differential equations and its integrability, Lett. Math. Phys. 3 (1979), 489.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Infinite conformal symmetry in two dimensional quantum field theory, Nucl. Phys. B241 (1984), 333.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. J.L. Gervais, A. Neveu, Dimension shifting operators and null states in 2D conformally invariant field theories, Nucl. Phys. B264 (1986), 557.

    CrossRef  MathSciNet  Google Scholar 

  12. E.Sklyanin, On the complete integrability of the Landau-Lifshitz equation, Preprint LOMI E-3-79, Leningrad.

    Google Scholar 

  13. O.Babelon, D.Bernard, Dressing Transformations and the Origin of the Quantum Group Symmetries, Preprint SPhT 91/016, PAR LPTHE 91/15.

    Google Scholar 

  14. M. Semenov-Tian-Shansky, Dressing Transformations and Poisson Lie Group Actions, Pub. RIMS 21 (1985), 1237.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. O.Babelon, L.Bonora, F.Toppan, Exchange algebra and the Drinfeld-Sokolov theorem, Preprint SISSA 65/90/EP; Commun. Math. Phys., (to appear).

    Google Scholar 

  16. O. Babelon, L. Bonora, Quantum Toda Theory, Phys. Lett. 253B (1991), 365.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. L.D.Faddeev, Integrable Models in 1+1 Dimensional Quantum Field Theory, Les Houches Lectures, 1982; Elsevier, Amsterdam, 1984.

    Google Scholar 

  18. M. Jimbo, A q-Difference analogue of μ(G) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. V.G.Drinfeld, Quantum Groups, Proc. of the ICM-86, vol. 1, Berkeley, 1987, p. 798.

    Google Scholar 

  20. O.Babelon, Universal exchange algebra for Bloch waves and Liouville theory, Preprint PAR LPTHE 91/11; Commun. Math. Phys., (to appear).

    Google Scholar 

  21. J.L. Geryais, The quantum group structure of 2D gravity and minimal models, Commun. Math. Phys. 130 (1990), 257.

    CrossRef  Google Scholar 

  22. J.L. Gervais, Solving the strongly coupled 2D gravity: 1. Unitary truncation and quantum group structure, Preprint LPTENS 90/13.

    Google Scholar 

  23. A. Volkov, Zapiski Nauch. Semin. LOMI 150 (1986), 17; Zapiski Nauch. Semin. LOMI 151 (1987), 24, (To be translated in Sov. Jour. Math.).

    Google Scholar 

  24. A. Volkov, Miura transformation on the lattice, Theor. Math. Phys. 74 (1988), 135.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. O. Babelon, Exchange formula and lattice deformation of the Virasoro algebra, Phys. Lett. 238B (1990), 234.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. O.Babelon, Integrable systems associated to the lattice version of the Virasoro algebra. I. The classical open chain, Talk given at the Workshop “Integrable systems and quantum groups” Pavia March 1–2 1990. Preprint PAR LPTHE 90/5.

    Google Scholar 

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© 1992 Springer-Verlag

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Babelon, O. (1992). Liouville theory on the lattice and universal exchange algebra for bloch waves. In: Kulish, P.P. (eds) Quantum Groups. Lecture Notes in Mathematics, vol 1510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101188

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  • DOI: https://doi.org/10.1007/BFb0101188

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55305-2

  • Online ISBN: 978-3-540-47020-5

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