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Lie algebras of differential operators, their central extensions, and W-algebras

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Literature Cited

  1. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, "Infinite conformal symmetry in two-dimensional quantum field theory," Nucl. Phys.,B241, 333–380 (1984).

    Google Scholar 

  2. G. Moore and N. Seiberg, "Classical and quantum conformal field theory," Commun. Math. Phys.,123, No. 2, 177–254 (1989).

    Google Scholar 

  3. A. B. Zamolodchikov, "Infinite additional symmetries in two-dimensional conformal quantum field theory," Teor. Mat. Fiz.,65, 347–359 (1985).

    Google Scholar 

  4. S. L. Luk'yanov, "Quantization of the Gel'fand—Dikii bracket," Funkts. Anal. Prilozhen.,22, No. 4, 1–10 (1988).

    Google Scholar 

  5. S. L. Luk'yanov and V. A. Fateev, "Conformally invariant models of two-dimensional quantum field theory with Zn-symmetry," Zh. Éksp. Teor. Fiz.,94, No. 3, 23–37 (1988).

    Google Scholar 

  6. V. G. Drinfel'd and V. V. Sokolov, "Lie algebras and equations of Korteweg—de Vries type," J. Sov. Math.,30, No. 2, 1975–2036 (1985).

    Google Scholar 

  7. A. A. Belavin, Report in Proc. of 2nd Yukawa Memorial Symp., Nishinomiya, Japan (1987), Ser. Proc. in Phys.,31, 132 (1989).

    Google Scholar 

  8. M. A. Bershadskii and H. Ooguri, "Quantum Hamiltonian reduction and W-algebras," Preprint IASSNS/HEP 1989/09 (1989).

  9. I. Bakas, "The large-N limit of extended conformal symmetries," Phys. Lett.,B228, 57–63 (1989).

    Google Scholar 

  10. A. Yu. Morozov, "On the concept of universal W-algebra," ITEP Preprint, ITEP 148/89 (1989).

  11. I. M. Gel'fand and L. A. Dikii, "A family of Hamiltonian structures for nonlinear integrable equations," in: I. M. Gel'fand — Collected papers. Vol. 1, Springer-Verlag, Berlin (1989), pp. 625–641.

    Google Scholar 

  12. T. G. Khovanova, "The Gel'fand—Dikii algebras and the Virasoro algebra," Funkts. Anal. Prilozh.,20, No. 4, 89–90 (1986).

    Google Scholar 

  13. A. O. Radul, "A central extension of the Lie algebra of differential operators on the circle and W-algebras," Pis'ma Zh. Éksp. Teor. Fiz.,50, No. 8, 341–343 (1989).

    Google Scholar 

  14. A. O. Radul, "Nontrivial central extensions of Lie algebras of differential operators in two and higher dimensions," Phys. Lett., to appear.

  15. O. S. Kravchenko and B. A. Khesin, "A nontrivial central extension of the Lie algebra of (pseudo-) differential symbols on the circle," Funkts. Anal. Prilozhen.,23, No. 3, 78–79 (1989).

    Google Scholar 

  16. J.-L. Brylinski and E. Getzier, "The homology of algebras of pseudodifferential symbols and the noncommutative residue," K-Theory,1, 385–403 (1987).

    Google Scholar 

  17. M. Wodzicki, "Noncommutative residue. Fundamentals," in: Lecture Notes in Math., Vol. 1289, Springer-Verlag, Berlin (1985), pp. 320–399.

    Google Scholar 

  18. Yu. I. Manin, "Neveu—Schwarz sheaves and differential equations for Mumford superforms," J. Geom. Phys.,5, No. 2, 161–181 (1988).

    Google Scholar 

  19. M. Adler, "On a trace functional for formal pseudodifferential operators and the symplectic structure of Korteweg—de Vries type equations," Inventiones Math.,50, 219–248 (1979).

    Google Scholar 

  20. Yu. I. Manin, "Algebraic aspects of nonlinear differential equations," in: Sov. Probl. Mat. Tom 11 (Itogi Nauki i Tekhniki VINITI A. N. SSSR), Moscow (1978), pp. 5–152.

    Google Scholar 

  21. M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  22. V. G. Drinfel'd, "Quantum groups," in: Proc. International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, R.I. (1987), pp. 798–820.

    Google Scholar 

  23. M. A. Semenov-Tyan-Shanskii, "What is a classical R-matrix?," Funkts. Anal. Prilozhen.,17, No. 4, 17–33 (1983).

    Google Scholar 

  24. M. A. Semenov-Tyan-Shanskii, "Dressing transformations and Poisson group actions," Publ. RIMS Kyoto Univ.,21, No. 6, 1237–1260 (1985).

    Google Scholar 

  25. V. I. Arnol'd and A. B. Givental’, "Symplectic geometry," in: Sov. Probl. Mat., Fundamental. Napravl., Tom 1 (Itogi Nauki i Tekhniki VINITI AN SSSR), Moscow (1985), pp. 7–149.

    Google Scholar 

  26. B. L. Feigin, "The Lie algebras gℓ (λ) and the cohomology of the Lie algebra of differential operators," Usp. Mat. Nauk,35, No. 2, 157–158 (1988).

    Google Scholar 

  27. A. Pressley and G. Segal, Loop Groups, The Clarendon Press, Oxford University Press, New York (1986).

    Google Scholar 

  28. D. A. Leites, "Theory of supermanifolds," Usp. Mat. Nauk,35, No. 1, 2–117 (1988).

    Google Scholar 

  29. Yu. I. Manin, Gauge Fields and Complex Geometry, Springer-Verlag, Berlin (1988).

    Google Scholar 

  30. Yu. I. Manin and A. O. Radul, "A supersymmetric extension of the Kadomtsev—Petviashvili hierarchy," Commun. Math. Phys.,98, 65–77 (1985).

    Google Scholar 

  31. D. A. Leites and B. L. Feigin, "New Lie superalgebras of string theory," in: Teoretiko-Gruppovye Metody v Fizike, Tom 1, Nauka, Moscow (1983), pp. 269–278.

    Google Scholar 

  32. A. Bilal, "A remark on N → ∞ limit of WN-algebras," Phys. Lett.,B227, 406–411 (1989).

    Google Scholar 

  33. C. N. Pope, L. J. Romans, and X. Shen, "The complete structure of W∞," Phys. Lett.,B236, 173–178 (1990).

    Google Scholar 

  34. I. Bakas and E. Kiritis, "On the structure of W-algebras," Preprint Univ. of Maryland, UMD-PP90-160 (1990).

  35. J.-L. Gervais, "W-algebras and two-dimensional Toda equations," Phys. Lett.,B160, 277–284 (1985).

    Google Scholar 

  36. A. V. Marshakov and A. Morozov, "Note on W3-algebra," ITEP Preprint, ITEP/146-89 (1989).

  37. A. A. Beilinson, Yu. I. Manin, and V. V. Shekhtman, "Sheaves of the Virasoro and Neveu—Schwarz algebras," in: Lecture Notes in Math., Vol. 1289, Springer-Verlag, Berlin (1987), pp. 52–71.

    Google Scholar 

  38. V. G. Kac and P. Peterson, "Representations of infinite-dimensional Lie algebras," Preprint MIT 27/87 (1987).

  39. A. O. Radul, "Superanalogues of Schwarz derivative and Bott cocycle," in: Seminar on Supermanifolds, Univ. of Stockholm, Vol. 21 (1986), pp. 40–57.

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V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 1, pp. 33–49, January–March, 1991.

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Radul, A.O. Lie algebras of differential operators, their central extensions, and W-algebras. Funct Anal Its Appl 25, 25–39 (1991). https://doi.org/10.1007/BF01090674

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