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Lectures on Classical W-Algebras

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These are lecture notes of lectures given in 1993 in Cortona, Italy. W-algebras appeared in the conformal field theory as extensions of the Virasoro algebra. They are closely connected with integrable systems and can be interpreted as symplectic (or Poisson) structures inherent in those systems (we touch only classical aspects of W-algeabras). We try to give some self-contained description of this link. First three parts are devoted to integrable systems and their Hamiltonian nature. Part 4 deals with the most important for W-algebras problem: seeking generators that are primary fields. Parts 4 and 5 introduce the KP-hierarchy and τ-functions, Part 6 discusses additional symmetries, W∞-algebra and the string equation. Finally, Part 7 is devoted to some reductions (constrained hierarchies).

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References

  1. Belavin, A. A., Polyakov, A. M., and Zamolodchikov A. B.: Infinite conformal symmetry in two dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333.

    Google Scholar 

  2. Gelfand, I. M. and Fuks, D. B.: Cohomologies of the Lie algebra of the vector fields on the circle, Funct. Anal. Appl. 2(4) (1968), 92–93.

    Google Scholar 

  3. Bouwknegt, P. and Schoutens, K.: W symmetry in conformal field theory, Phys. Reports 223(4) (1993), 183–286.

    Google Scholar 

  4. Zamolodchikov, A. B.: Infinite additional symmetries in two-dimensional conformal quantum field theory, Theoret. Math. Phys. 65 (1985), 1205.

    Google Scholar 

  5. Fateev, V. A. and Lukyanov, S.L.: Additional symmetries and exactly solvable models of two dimensional conformal field theory, Internat. J. Modern Phys. A3 (1988), 507.

    Google Scholar 

  6. Lukyanov, S. L.: Quantization of the Gelfand-Dikey bracket, Funct. Anal. Appl. 22 (1990), 1.

    Google Scholar 

  7. Gervais, G.-L.: Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson brackets, Phys. Lett. B 160 (1985), 277.

    Google Scholar 

  8. Gardner, C. S.: Korteweg-de Vries equation and generalizations IV, J. Math. Phys. 12(8) (1971), 1548–1551.

    Google Scholar 

  9. Zakharov, V. E. and Faddeev, L. D.: The Korteweg-de Vries equation is a completely integrable Hamiltonian system, Funct. Anal. Appl. 5(4) (1971), 18–27.

    Google Scholar 

  10. Magri, F.: A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19(5) (1978), 1156–1162.

    Google Scholar 

  11. Gelfand, I. M. and Dickey, L. A.: Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), 259–273.

    Google Scholar 

  12. Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equations, Invent. Math. 50(3) (1979), 219–248.

    Google Scholar 

  13. Gelfand, I. M. and Dickey, L. A.: Family of Hamiltonian structures connected with integrable non-linear differential equations, Preprint, IPM, Moscow (in Russian), 1978.

    Google Scholar 

  14. Dickey, L. A.: Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991.

    Google Scholar 

  15. Gelfand, I. M. and Dickey, L. A.: Family of Hamiltonian structures connected with integrable nonlinear differential equations, Preprint IPM, (1978), 1–136.

  16. Dickey, L. A.: Soliton Equations and Hamiltonian Systems, Adv. Series of Math. Phys. 12, World Scientific, Singapore, 1991.

    Google Scholar 

  17. Kupershmidt, B. A. and Wilson, G.: Modifying Lax equations and the second Hamiltonian structure, Invent. Math. 62 (1982), 403–436.

    Google Scholar 

  18. Dickey, L. A.: A short proof of a Kupershmidt and Wilson theorem, Comm. Math. Phys. 87 (1983), 127–129.

    Google Scholar 

  19. Drinfeld, V. G. and Sokolov, V. V.: Equations of Korteweg-de Vries type and simple Lie algebras, Dokl. AN USSR 258(1) (1981), 11–16.

    Google Scholar 

  20. Dickey, L. A.: Integrable nonlinear equations and Liouville's theorem II, Comm. Math. Phys. 82 (1981), 361–375.

    Google Scholar 

  21. Di Francesco, P., Itzykson, C., and Zuber, J.-B.: Classical W-algebras, Comm. Math. Phys. 140 (1991), 543–567.

    Google Scholar 

  22. Radul, A. O.: Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. Appl. 25(1) (1991).

  23. Bakas, I.: Higher spin fields and the Gelfand-Dickey algebra, Comm. Math. Phys. 123 (1989), 627–639.

    Google Scholar 

  24. Balog, J., Fehér, L., O'Raifeartaigh, L., Forgács, P., and Wipf, A.: Toda theory and W-algebra from a gauged WZNW point of view, Ann. of Phys. 203(1) (1990), 76–136.

    Google Scholar 

  25. Bauer, M., Di Francesco, P., Itzykson, C., and Zuber, J.-B.: Covariant differential equations and singular vectors in Virasoro representations, Nuclear Phys. B 362 (1991), 515–562.

    Google Scholar 

  26. Bonora, L. and Xiong, C. S.: Covariant sl 2 decomposition of the sl n Drinfeld-Sokolov equations and the W n algebras, Internat. J. Modern Phys. A 7(7) (1992), 1507–1525.

    Google Scholar 

  27. Figueroa-O'Farrill, J. M. and Ramos, E.: Existence and uniqueness of the universal W-algebra, J. Math. Phys. 33(3) (1992), 833–839.

    Google Scholar 

  28. Figueroa-O'Farrill, J. M., Mas, J., and Ramos, E.: The topography of W ∞-type algebras, Preprint, hep-th/9208077 (1992).

  29. Bakas, I.: The sructure of the W ∞ algebra, Comm. Math. Phys. 134 (1990), 487–508.

    Google Scholar 

  30. Watanabe, Y.: Hamiltonian structure of Sato's hierarchy of KP equations and a coadjoint orbit of a certain formal Lie group, Lett. Math. Phys. 7(2), (1983), 99–106.

    Google Scholar 

  31. Dickey, L.A.: On Hamiltonian and Lagrangian formalisms for the KP hierarchy of integrable equations, Ann. N.Y. Acad. Sci. 491 (1987), 131–148.

    Google Scholar 

  32. Radul, A.O.: Two series of Hamiltonian structures for the hierarchy of Kadomtsev—Petviashvili equations, in Mironov, Moroz and Tshernyatin (eds), Applied Methods of Nonlinear Analysis and Control, Moscow, State Univ., 1987, pp. 149–157.

    Google Scholar 

  33. Date, E., Jimbo, M., Kashiwara, M., and Miwa, T.: Transformation groups for soliton equations, in M. Jimbo and T. Miwa (eds), Non-Linear Integrable Systems — Classical Theory and Quantum Theory, Proc. RIMS Symposium, Singapore, 1983.

    Google Scholar 

  34. Kac, V. G. and Raina, A. K.: Highest Weight Representations of Infinite-Dimensional Lie Algebras, Adv. Series Math. Phys. 2, World Scientific, Singapore, 1987.

    Google Scholar 

  35. Dickey, L. A.: Soliton Equations and Hamiltonian Systems, Adv. Series in Math. Phys. 12, World Scientific, Singapore, 1991.

    Google Scholar 

  36. Van Moerbeke, P.: Integrable foundations of string theory, in O. Babilon, P. Cartier, and Y. Kosmann-Schwarzbach (eds), Lectures on Integrable Systems, World Scientific, Singapore, 1994.

    Google Scholar 

  37. Orlov, A. Yu. and Shulman, E. I.: Additional symmetries for integrable and conformal algebra representation, Lett. Math. Phys. 12 (1986), 171.

    Google Scholar 

  38. Adler, M., Shiota, T., and Van Moerbeke, P.: From the w ∞-algebra to its central extension: a τ-function approach, Phys. Lett. A 194 (1994), 33–43.

    Google Scholar 

  39. Adler, M. and Van Moerbeke, P.: A matrix solution to two-dimensional W p-gravity, Comm. Math. Phys. 147 (1992), 25–26.

    Google Scholar 

  40. Aratyn, H., Nissimov, E., and Pacheva, S.: Construction of KP hierarchies in terms of finite number of fields and their abelianization, Preprint, hep-th/9306035 (1993).

  41. Bonora, L. and Xiong, C. S.: The (N, M)th KdV hierarchy and the associated W algebra, Preprint, SISSA, hep-th/9311070 (1993). Bonora, L., Liu, Q. P., and Xiong, C. S.: The integrable hierarchy constructed from a pair of higher KdV hierarchies and its associated W algebra, Preprint, hep-th/9408035 (1994).

  42. Dickey, L. A.: On the constrained KP hierarchy II, Lett. Math. Phys. 35(3) (1995), 229–236 (hep-th/9411005, 9411157).

    Google Scholar 

  43. Fehér, L., Harnad, I., and Marshall, I.: Generalized Drinfeld-Sokolov reductions and KdV type hierarchies, Comm. Math. Phys. 154 (1993), 181–214. Fehér, L. and Marshall, I.: Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction, Preprint, hep-th/9503217 (1995).

    Google Scholar 

  44. Dickey, L. A.: Soliton Equations and Integrable Systems, Adv. Series in Math. Phys. 12, World Scientific, Singapore, 1991.

    Google Scholar 

  45. Yu, Feng: Many boson realizations of universal nonlinear W ∞ algebras, Lett. Math. Phys. 29 (1993), 175 (hep-th/9301053).

    Google Scholar 

  46. Oevel, W. and Strampp, W.: Constrained KP hierarchy and bi-Hamiltonian structures, Comm. Math. Phys. 157 (1993), 51–81.

    Google Scholar 

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  1. Department of Mathematics, University of Oklahoma, Norman, OK, 73019, U.S.A.

    L. A. Dickey

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Dickey, L.A. Lectures on Classical W-Algebras. Acta Applicandae Mathematicae 47, 243–321 (1997). https://doi.org/10.1023/A:1017903416906

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