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Coset algebras, integrable hierarchies and matrix models

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References

  1. Ablowitz M.J., Clarkson P. A. Solitons, nonlinear evolution equations and inverse scattering, Lond. Math. Soc. Lect. Note Ser. 149. Cambridge University Press, 1991.

  2. Ahn C., Shigemoto K. Multi-matrix model and 2D Toda multi-component hierarchy, Phys. Lett. 1991. Vol. 263B. pp. 44–50.

    Google Scholar 

  3. Altschüler D., Beran P., Lacki J., Roditi I. Twisted vertex operators and representations of finite Heisenberg groups, Int. J. Mod. Phys. 1989. Vol. A4, pp. 921–942.

    Google Scholar 

  4. Alvarez-Gaumé L., Gomez C., Lacki J. Integrability in random matrix models, Phys. Lett. 1991. Vol. 253B. pp. 56–62.

    Google Scholar 

  5. Ambjørn J., Durhuus B., Fröhlich J. Diseases of triangulated random surface models, and possible cures, Nucl. Phys. 1985. Vol. B257. [FS14]. pp. 433–449.

    Google Scholar 

  6. Anderson G., Moore G. Rationality in conformal field theory, Comm. Math. Phys. 1988. Vol. 117. pp. 441–450.

    Google Scholar 

  7. Bagger J. Basic conformal field theory. Lectures presented at the 1988 Banff Summer Institute on Particles and Fields.

  8. Bagger J., Nemeschansky D. Coset construction of chiral algebras. Harvard preprint HUTP-88/A059, published in ‘College Park Workshop 1988’.

  9. Bais F. A., Bouwknegt P., Schoutens K., Surridge M. Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants, Nucl. Phys. 1988, Vol. B304. pp. 348–370.

    Google Scholar 

  10. Bais F. A., Bouwknegt P., Schoutens K., Surridge M. Coset construction for extended Virasoro algebras, Nucl. Phys. 1988. Vol. B304. pp. 371–391.

    Google Scholar 

  11. Bais F. A., Englert F., Taormina A., Zizzi P. Torus compactification for non-simply laced groups, Nucl. Phys. 1987. Vol. B279. pp. 529–547.

    Google Scholar 

  12. Bais F. A., de Vos K. A coset construction for integrable hierarchies, in ‘Differential Geometric Methods in Theoretical Physics’. Eds. Chau L.-L., Nahm W. Proc. Conf. Lake Tahoe, California, 1988. Plenum Press: New York, 1990. pp. 291–296; Preprint ITFA 89-28, Amsterdam, 1989.

    Google Scholar 

  13. Bais F. A., de Vos K. The algebraic structure of integrable hierarchies in bi-linear form, in ‘Recent Developments in Conformal Field Theories’. Eds. Randjbar-Daemi S., Sezgin E., Zuber J.-B. Proc. Trieste Conf. Italy, 1989. World Scientific, Singapore, 1990. pp. 271–279; Preprint ITFA 90-04, Amsterdam, 1990.

    Google Scholar 

  14. Bais F. A., Taormina A. Accidental degeneracies in string compactification, Phys. Lett. 1986. Vol. 181B. pp. 87–90.

    Google Scholar 

  15. Bais F. A., Tjin T., van Driel P. Covariantly coupled chiral algebras, Nucl. Phys. 1991. Vol. B357. pp. 632–654.

    Google Scholar 

  16. Bakas I. The Hamiltonian structure of the spin 4 operator algebra, Phys. Lett. 1988. Vol. 213B. pp. 313–318; Higher spin fields and the Gelfand-Dickey algebras, Comm. Math. Phys. 1989. Vol. 123. pp. 627–639.

    Google Scholar 

  17. Bardacki K., Halpern M. B. New dual quark models, Phys. Rev. 1971. Vol. D3. pp. 2493–2506.

    Google Scholar 

  18. Barut A. O., Raczka R. Theory of group representations and applications. World Scientific, 1986.

  19. Belavin A. A., Polyakov A. M., Zamolodchikov A. B. Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. 1984. Vol. B241. pp. 333–380.

    Google Scholar 

  20. Bessis D., Itzykson C., Zuber J.-B. Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1980. Vol. 1. pp. 109–157.

    Google Scholar 

  21. Block B., Yankielowicz S. Extended current algebras and the coset construction of conformal field theories, Nucl. Phys. 1989. Vol. B315. pp. 25–42.

    Google Scholar 

  22. Blumenhagen R., Flohr M., Kliem A., Nahm W., Recknagel A., Varnhagen R. W-algebras with two and three generators, Nucl. Phys. 1991. Vol. B361. pp. 255–289.

    Google Scholar 

  23. Bouwknegt P. On the construction of modular invariant partition functions, Nucl. Phys. 1987. Vol. B290 [FS20]. pp. 507–526.

    Google Scholar 

  24. Bouwknegt P. Extended conformal algebras, Phys. Lett. 1988. Vol. 270B. p. 295.

    Google Scholar 

  25. Bouwknegt P. Extended conformal algebras from Kac-Moody algebras, in ‘Infinite-dimensional Lie algebras and Lie groups’. Ed. Kac. V. G.. Proc. CIRM-Luminy conf. 1988. World Scientific. Singapore, 1989.

    Google Scholar 

  26. Bowcock P. Quasi-primary fields and associativity of chiral algebras, Nucl. Phys. 1991. Vol. B356. pp. 367–386.

    Google Scholar 

  27. Bowcock P., Goddard P. Coset constructions and extended conformal algebras, Nucl. Phys. 1988. Vol. B305. pp. 685–709.

    Google Scholar 

  28. Bowcock P., Watts G. M. T. On the classification of quantumW algebras, Nucl. Phys. 1992. Vol. B379. pp. 63–95.

    Google Scholar 

  29. Bowick M. J., Morozov A., Shevitz D. Reduced unitary matrix models and the hierarchy of τ-functions, Nucl. Phys. 1991. Vol. B354. pp. 496–530.

    Google Scholar 

  30. Brézin E., Itzykson C., Parisi G., Zuber J.-B. Planar diagrams, Comm. Math. Phys. 1978. Vol. 59. pp. 35–51.

    Google Scholar 

  31. Brézin E., Kazakov V. Exactly solvable field theories of closed strings, Phys. Lett. 1990. Vol. 236B. pp. 144–150.

    Google Scholar 

  32. Cappelli A., Itzykson C., Zuber J.-B. The A-D-E Classification of Minimal andA (1)1 Conformal Invariant Theories, Comm. Math. Phys. 1987. Vol. 113. pp. 1–26.

    Google Scholar 

  33. Cardy J. L. Conformal invariance, in ‘Phase transitions and critical phenomena’. Vol. 11. Eds. Domb C., Lebowitz J. Academic Press, 1986.

  34. Cardy J. L. Operator content of two-dimensional conformally invariant theories, Nucl. Phys. 1986. Vol. B270 [FS16]. pp. 186–204.

    Google Scholar 

  35. Cvitanović P. Group theory. Nordita notes, 1984.

  36. Damour T., Taylor J. H. Strong-field tests of relativistic gravity and binary pulsars, Phys. Rev. D. 1992. Vol. 45. pp. 1840–1868.

    Google Scholar 

  37. Das A. Integrable models. Lecture Notes in Physics. Vol. 30. World Scientific Singapore, 1989.

    Google Scholar 

  38. Date E., Jimbo M., Kashiwara M., Miwa T. Transformation groups for soliton equations, Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci. 1982. Vol. 18. pp. 1077–1110.

    Google Scholar 

  39. Date E., Kashiwara M., Miwa T. Vertex operators and τ-functions, Proc. Japan Acad. 1981. Vol. 57A. pp. 387–392.

    Google Scholar 

  40. David F. Planar diagrams, two-dimensional lattice gravity and surface models. A model of random surfaces with non-trivial critical behaviour, Nucl. Phys. 1985. Vol. B257 [FS14]. pp. 45–58, 543–576.

    Google Scholar 

  41. De Bakker B. V. W-algebras in the coset modelsu(2)/u(1). Masters thesis, Amsterdam. 1991.

  42. De Boer J. Multimatrix models and the KP-hierarchy, Nucl. Phys. 1991. Vol. B366. pp. 602–628.

    Google Scholar 

  43. Dengyuan C., Hangwei Z. Lie algebraic structure for the AKNS system, J. Phys. 1991. Vol. A24. pp. 377–383.

    Google Scholar 

  44. Dickey L. A. Soliton equations and Hamiltonian systems. Adv. Ser. Math. Phys., Vol. 12. World Scientific. Singapore, 1991.

    Google Scholar 

  45. Dijkgraaf R., Verlinde E. Modular invariance and the fusion algebra, Nucl. Phys. 1988. Vol. B. (Proc. Suppl.) Vol. 5B. pp. 87–97.

    Google Scholar 

  46. Dijkgraaf R., Verlinde H., Verlinde E. Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity, Nucl. Phys. 1991. Vol. B348. pp. 435–456.

    Google Scholar 

  47. Distler J., Kawai H. Conformal field theory and 2d quantum gravity, Nucl. Phys. 1989. Vol. B321. pp. 509–527.

    Google Scholar 

  48. Douglas M. R. Strings in less than one dimension and the generalized KdV hierarchies, Phys. Lett. 1990. Vol. 238B. pp. 176–180.

    Google Scholar 

  49. Douglas M., Shenker S. Strings in less than one dimension, Nucl. Phys. 1990. Vol. B335. pp. 635–654.

    Google Scholar 

  50. Drinfeld V. G., Sokolov V. V. Lie algebras and equations of KdV type, J. Sov. Math. 1985. Vol. 30. pp. 1975–2036.

    Google Scholar 

  51. Dynkin E. B. Semisimple subalgebras of semisimple Lie algebras, Maximal subgroups of the classical groups, Amer. Math. Soc. Transl. (2). 1957. Vol. 6. pp. 111–244, 245–378.

    Google Scholar 

  52. Fateev V. A., Lykyanov S. L. The models of two-dimensional conformal quantum field theory withZ n symmetry, Int. J. Mod. Phys. 1988. Vol. A3. pp. 507–520.

    Google Scholar 

  53. Fateev V. A., lykyanov S. L. Additional symmetries and exactly solvable models in two-dimensional conformal field theory I,II,III. Preprint I,II,III Landau Institute for Theoretical Physics. 1988.

  54. Fateev V., Zamolodchikov A. Conformal quantum field theory models in two dimensions havingZ 3 symmetry, Nucl. Phys. 1987. Vol. B280 [FS18]. pp. 644–660.

    Google Scholar 

  55. Fehér L., O'Raifeartaigh L., Ruelle P., Tsutsui I., Wipf A. On the general structure of Hamiltonian reductions of the WZWNW theory. Dublin preprint DIAS-STP-91-29, to appear in Phys. Rep.

  56. Fermi A., Pasta J., Ulam S. Studies of nonlinear problems I. Los Alamos report LA1940 (1955), in ‘Nonlinear Wave Motion’. Ed. Newell A. C. Lect. Appl. Math. Vol. 15. A.M.S., Providence, RI. 1974. pp. 143–196.

    Google Scholar 

  57. Friedan D., Martinec E., Shenker S. Conformal invariance, superymmetry, and string theory, Nucl. Phys. 1986. Vol. B271. pp. 93–165.

    Google Scholar 

  58. Friedan D., Qiu Z., Shenker S. Conformal invariance, unitarity, and critical exponents in two dimensions, Phys. Rev. Lett. 1984. Vol. 52 pp. 1575–1578; in ‘Vertex operators in mathematics and physics’. Eds. Lepowsky J. et al. Springer Verlag, 1984; Comm. Math. Phys. 1986. Vol. 107. pp. 535–542.

    Google Scholar 

  59. Friedan D., Shenker S. Unpublished, 1987.

  60. Fucito F., Martinelli M., Gamba A. Non-perturbative two-dimensional quantum gravity, KdV hierarchy and solutions of the string equations, Phys. Lett. 1990. Vol. 248B. pp. 57–61.

    Google Scholar 

  61. Fukuma M., Kawai H., Nakayama R. Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Int. J. Mod. Phys. 1991. Vol. A6. pp. 1385–1406.

    Google Scholar 

  62. Fukuma M., Kawai H., Nakayama R. Infinite Dimensional Grassmannian Structure of Two-Dimensional Quantum Gravity. Preprint UT-572, KEK-TH-272, KEK 90-165. 1990.

  63. Gardner C. S. The Korteweg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys. 1971. Vol. 12. pp. 1548–1551.

    Google Scholar 

  64. Gardner C. S., Greene J. M., Kruskal M. D., Miura R. M. Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 1967. Vol. 19. pp. 1095–1097.

    Google Scholar 

  65. Gelfand I. M., Dickey L. A. Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 1976. Vol. 10:4. pp. 13–29; The resolvent and Hamiltonian systems, Funct. Anal. Appl. 1977. Vol. 11:2. pp. 93–104.

    Google Scholar 

  66. Gepner D. On the spectrum of conformal field theories, Nucl. Phys. 1987. Vol. B287. pp. 111–130.

    Google Scholar 

  67. Gerasimov A., Marshakov A., Mironov A., Morozov A., Orlov A. Matrix models of two-dimensional gravity and Toda theory, Nucl. Phys. 1991. Vol. B357. pp. 565–618.

    Google Scholar 

  68. Gervais J. L. Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson brackets, Phys. Lett. 1985. Vol. 160B. pp. 277–278.

    Google Scholar 

  69. Gervais J. L., Neveu A. Dual string spectrum in Polyakov's quantization (II), Nucl. Phys. 1982. Vol. B209. pp. 125–145.

    Google Scholar 

  70. Ginsparg P. Applied conformal field theory, in Les Houches, Session XLIX, 1988, ‘Fields, strings and critical phenomena’. Eds. Brézin E., Zinn-Justin J. Elsevier, 1989.

  71. Ginsparg P. Matrix models of 2d gravity, Lectures given at 1991 Trieste summer school, Los Alamos preprint LA-UR-91-4101. December 1991.

  72. Goddard P. Meromorphic conformal field theory, in ‘Infinite-dimensional Lie algebras and Lie groups’. Ed. Kac V. G. Proc. CIRM-Luminy conf., 1988. World Scientific, Singapore, 1989.

    Google Scholar 

  73. Goddard P., Kent A., Olive D. Virasoro algebras and coset space models, Phys. Lett. 1985. Vol. 152B. pp. 88–92.

    Google Scholar 

  74. Goddard P., Nahm W., Olive D. Symmetric spaces, Sugawara's energy-momentum tensor in two dimensions and free ferminons, Phys. Lett. 1985. Vol. 160B. pp. 111–116.

    Google Scholar 

  75. Goddard P., Olive D. Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. 1986. Vol. A1. pp. 303–414.

    Google Scholar 

  76. Goeree J. W-constraints in 2D quantum gravity, Nucl. Phys. 1991. Vol. B358. pp. 737–757.

    Google Scholar 

  77. Green M. B., Schwarz J. H., Witten E. Superstring Theory. vol. 1 & 2. Cambridge University Press, 1987.

  78. Gross D., Migdal A. A. Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 1990. Vol. 64. pp. 127–130.

    Google Scholar 

  79. Gross D., Periwal V. String perturbation theory diverges, Phys. Rev. Lett. 1988. Vol. 60. pp. 2105–2108.

    Google Scholar 

  80. Halpern M. B., Kiritsis E., Mod. Phys. Lett. 1989. Vol. A4. pp. 1373–1380;Halpern M. B., Kiritsis E., Obers N. A., Porrati M., Yamron J. P. New unitary affine Virásoro constructions, Int. J. Mod. Phys. 1990. Vol. A5. p. 2275.

    Google Scholar 

  81. Hirota R. Direct methods in soliton theory, in Topics in Current Physics Vol. 17. Eds. Bullough R., Caudrey P. Springer-Verlag, New York, 1980, pp. 157–175.

    Google Scholar 

  82. Ho-Kim Q., Zheng H.B., Twisted conformal field theories withZ 3 invariance, Phys. Lett. 1988. Vol. 212B. pp. 71–76; Twisted structures of extended Virasoro algebras, Phys. Lett. 1989. Vol. 223B. pp. 57–60; Twisted characters and partition functions in extended Virasoro algebras, Mod. Phys. Lett. 1990. Vol. A5. pp. 1181–1189.

    Google Scholar 

  83. Itoyama H. Matsuo Y. Noncritical Virasoro algebra of thed<1 matrix model and the quantized string field, Phys. Lett. 1991. Vol. 255B. pp. 202–208.

    Google Scholar 

  84. Itoyama H., Matsuo Y. ω1+∞-type constraints in matrix models at finiteN, Phys. Lett. 1991. Vol. 262B. pp. 233–239.

    Google Scholar 

  85. Kac V.G. Simple graded Lie algebras of finite growth, Funct. Anal. Appl. 1967. Vol. 1. pp. 328–329.

    Google Scholar 

  86. Kac V.G. Infinite dimensional Lie algebras. Cambridge: Cambridge University Press, 1985.

    Google Scholar 

  87. Kac V.G. Automorphisms of finite order of semi-simple Lie algebras, Funct. Anal. Appl. 1969. Vol. 3. pp. 252–254;Helgason S. Differential geometry, Lie groups and symmetric spaces. Academic Press, 1978.

    Google Scholar 

  88. Kac V.G., Peterson, D.H. 112 constructions of the basic representation of the loop group ofE 8, Proc. Conf. on Anomalies, Geometry and Topology. Argonne, Chicago. Ed. Bardeen W.A., White A.R. World Scientific, Singapore, 1985. pp. 276–298.

    Google Scholar 

  89. Kac V.G., Raina A.K. Highest weight representations of infinite dimensional Lie algebras. Adv. Ser. in Math. Phys. Vol. 2. World Scientific, Singapore, 1987.

    Google Scholar 

  90. Kac V.G., Schwarz A. Geometric interpretation of partition function of 2D-gravity, Phys. Lett. 1991. Vol. 258B. pp. 329–334.

    Google Scholar 

  91. Kac M., van Moerbeke P., Proc. Natl. Acad. Sci. USA. 1975. Vol. 72. p. 1627.

    Google Scholar 

  92. Kac V.G., Wakimoto M. Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 1988. Vol. 70. pp. 156–236.

    Google Scholar 

  93. Kac V.G., Wakimoto M. Exceptional hierarchies of solition equations, Proc. Symp. Pure Math. 1989. Vol. 49. Part 1. pp. 191–237.

    Google Scholar 

  94. Kadomtsev B.B., Petviashvili V.I. On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 1970. Vol. 31. pp. 539–541.

    Google Scholar 

  95. Kaku M. Strings, Conformal Fields, and Topology. An Introduction New York: Springer-Verlag, 1991.

    Google Scholar 

  96. Kato A. Classification of modular invariant partition functions in two dimensions, Mod. Phys. Lett. 1987. Vol. A2. p. 585.

    Google Scholar 

  97. Kausch H.G., Watts G.M.T. A study ofW-algebras using Jacobi identities, Nucl. Phys. 1991. Vol. B354. pp. 740–768.

    Google Scholar 

  98. Kazakov V.G., Kostov I.K., Migdal A.A. Critical properties of randomly triangulated planar random surfaces, Phys. Lett. 1985. Vol. 157B. p. 295.

    Google Scholar 

  99. Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B. Fractal structure of 2D quantum gravity, Mod. Phys. Lett. 1988. Vol. A3. p. 819.

    Google Scholar 

  100. Korteweg D.J., de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. Ser. 5. 1895. Vol. 39. pp. 422–443.

    Google Scholar 

  101. La H.S. Symmetries in nonperturbative 2-d quantum gravity. Mod. Phys. Lett. 1991. Vol. A6. pp. 573–580; Geometry of Virasoro Constraints in Nonperturbative 2-d Quantum Gravity. Preprint UPR-0453T. 1990.

    Google Scholar 

  102. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 1968. Vol. 21. pp. 467–490.

    Google Scholar 

  103. Lepowsky J., Wilson R.L. Construction of the affine Lie algebraA 1 (1), Comm. Math. Phys. 1978 Vol. 62. pp. 43–53.

    Google Scholar 

  104. Magri F. A simple model of the integrable Hamiltonian equation, J. Math. Phys. 1978. Vol. 19. pp. 1156–1162.

    Google Scholar 

  105. Makeenko Yu., Marshakov A., Mironov A., Morozov A. Continuum versus discrete Virasoro in onematrix models, Nucl. Phys. 1991. Vol. B356. pp. 574–628.

    Google Scholar 

  106. Martinec E.J. On the origin of integrability in matrix models, Comm. Math. Phys. 1991. Vol. 138. pp. 437–449.

    Google Scholar 

  107. Mathieu P. Extended classical conformal algebras and the second Hamiltonian structure of Lax equations, Phys. Lett. 1988. Vol. B208. pp. 101–106.

    Google Scholar 

  108. Miura R.M. Korteweg-de Vries equations and generalizations I. A. remarkable explicit nonlinear transformation, J. Math. Phys. 1968. Vol. 9. pp. 1202–1204.

    Google Scholar 

  109. Moody R.V. Lie algebras associated with generalized Cartan matrices, Bull. Am. Math. Soc. 1967. Vol. 73. pp. 217–221.

    Google Scholar 

  110. Moore G., Seiberg N. Polynomial equations for rational conformal field theories, Phys. Lett. 1988. Vol. 212B. pp. 451–460.

    Google Scholar 

  111. Moore G., Seiberg N. Naturality in conformal field theory, Nucl. Phys. 1989. Vol. B313, pp. 16–40; Classical and quantum conformal field theory, Comm. Math. Phys. 1989. Vol. 123, pp. 177–254.

    Google Scholar 

  112. Nahm W. Talk given at the Conference on Recent Developments in Conformal Field Theory, Trieste. October, 1989.

  113. Nahm W. unpublished.

  114. Narganes Quijano F.J. On the parafermionicW N-algebra, Int. J. Mod. Phys. 1991. Vol. A6. pp. 2611–2634.

    Google Scholar 

  115. Newell A. C. Solitons in mathematics and physics. SIAM, 1985.

  116. Obers N. private communication.

  117. Orlov A. Yu., Schulman E.I. Additional symmetries for integrable equations and conformal algebra representation, Lett. Math. Phys. 1986. Vol. 12. pp. 171–179;Grinevich P. G., Orlov A. Yu. Virasoro action on Riemann surfaces, Grassmannians, det\(\bar{\partial}_J \) and Segal-Wilson τ-function, in ‘Problems of Modern Quantum Field Theory’. Eds. Belavin A.A. et al. Springer Verlag. Berlin, Heidelberg, 1989.

    Google Scholar 

  118. Peterson D.H., Kac V.G. Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA. 1983. Vol. 80. p. 1778.

    Google Scholar 

  119. Polyakov A. M. Quantum geometry of bosonic strings, Phys. Lett. 1981. Vol. 103B. pp. 207–210.

    Google Scholar 

  120. Polyakov A.M. Mod. Phys. Lett. 1987. Vol. A2. p. 893; Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. 1990. Vol. A5. p. 833.

    Google Scholar 

  121. Pope C.N., Romans L.J., Shen X. A brief history ofW ∞ in Proceedings of ‘Strings 90’, Texas A& M. Eds. Arnowitt R. et al. World Scientific, 1991.

  122. Post G. Pure Lie algebraic approach to the modified Korteweg-de Vries equation, J. Math. Phys. 1986. Vol. 27. pp. 678–681.

    Google Scholar 

  123. Scott Russell J. Report of the committee on waves, Report of the 7th Meeting of the British Association for the Advancement of Science, Liverpool, 1838. pp. 417–496.

  124. Sato M. Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku. 1981. Vol. 439. pp. 30–46.

    Google Scholar 

  125. Schoutens K. Extensions of conformal symmetry in 2-dimensional quantum field theory. Ph.D. thesis. University of Utrecht, 1989.

  126. Schrans S. Extensions of conformal invariance in two-dimensional quantum field theory. PhD thesis. University of Leuven, 1991.

  127. Schwarz A. On some mathematical problems of 2D-gravity andW h-gravity, Mod. Phys. Lett. 1991. Vol. A6. pp. 611–616.

    Google Scholar 

  128. Segal G., Wilson G. Loop groups and equations of KdV type, Publ. Math. I.H.E.S. 1985 Vol. 61. pp. 1–65.

    Google Scholar 

  129. Semikhatov A.M. Soliton equations and the Virasoro algebra, Int. J. Mod. Phys. 1989. Vol. A4. pp. 467–480.

    Google Scholar 

  130. Smoot G.F., Bennett C.L., Wright E.L., Kogut A. et al. To be published in Astrophys. J. Lett. 1992.

  131. Sugawara H. A field theory of currents, Phys. Rev. 1968. Vol. 170. pp. 1659–1662.

    Google Scholar 

  132. ten Kroode A.P.E., Bergvelt M.J. The homogeneous realization of the basic representation ofA 1 (1) and the Toda lattice, Lett. Math. Phys. 1986. Vol. 12. pp. 139–147.

    Google Scholar 

  133. Thielemans K.A. Mathematica package for computing operator product expansions, Int. J. Mod. Phys. 1991. Vol. C2. pp. 787–798.

    Google Scholar 

  134. 't Hooft G. A planar diagram theory for strong interactions, Nucl. Phys. 1974. Vol. B72. pp. 461–473.

    Google Scholar 

  135. Ueno K., Takasaki K. Toda lattice hierarchy, Adv. Studies in Pure Math. 1984. Vol. 4. p. 1.

    Google Scholar 

  136. Verlinde E. Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. 1988. Vol. B300. pp. 360–375.

    Google Scholar 

  137. Vermaseren J. FORM User's Guide. Nikhef-Amsterdam, 1990.

  138. Wakimoto M., Yamada H. Irreducible decompositions of Fock representations of the Virasoro algebra. Lett. Math. Phys. 1983. Vol. 7. pp. 513–516.

    Google Scholar 

  139. Watts G.M.T. W-algebras and coset models, Phys. Lett. 1990. Vol. 245B. pp. 65–71.

    Google Scholar 

  140. Watts G.M.T. Extended algebras in conformal field theory. PhD thesis, Trinity College. Cambridge, 1990.

    Google Scholar 

  141. Witten E. Non-abelian bosonization in two dimensions, Comm. Math. Phys. 1984. Vol. 92. pp. 455–472.

    Google Scholar 

  142. Witten E. Two dimensional gravity and intersection theory on moduli space. Princeton preprint IASSNS-HEP-90/45.

  143. Zabusky N.J., Kruskal M.D. Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 1965. Vol. 15. pp. 240–243.

    Google Scholar 

  144. Zakharov V.E., Faddeev L.D. The Korteweg-de Vries equation: a completely integrable Hamiltonian system, Funct. Anal. Appl. 1971. Vol. 5. pp. 280–287.

    Google Scholar 

  145. Zamolodchikov A.B. Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys. 1985. Vol. 65. pp. 1205–1213.

    Google Scholar 

  146. Zamolodchikov A.B., Zamolodchikov Al.B. Conformal field theory and critical phenomena, Sov. Sci. Rev. A. Phys. 1989. Vol. 10. pp. 269–433.

    Google Scholar 

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Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 2, pp. 163–287, May, 1993.

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de Vos, K. Coset algebras, integrable hierarchies and matrix models. Theor Math Phys 96, 879–973 (1993). https://doi.org/10.1007/BF01017700

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