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The principal realization of the basic representation. Application to the KdV-type hierarchies of non-linear partial differential equations

  • Victor G. Kac
Part of the Progress in Mathematics book series (PM, volume 44)

Abstract

The highest weight module L0) over an affine Lie algebra g(A) is called the basic representation of g(A). In this chapter we construct the basic representation explicitly in terms of certain (infinite order) differential operators in infinitely many indeterminates, called the vertex operators. The so-called principal Heisenberg subalgebra s of g(A) plays a crucial role in this construction. In a similar fashion, we construct representations of affine Lie algebras of infinite rank.

Keywords

Basic Representation Vertex Operator High Weight Vector Vacuum Vector Formal Completion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliographical notes and comments

  1. Kac, V. G. [ 1978 A] Infinite-dimensional algebras, Dedekind’s rI-finction, classical Möbious function and the very strange formula, Advances in Math. 30 (1978), 85–136.zbMATHCrossRefGoogle Scholar
  2. Kac, V. G., Kazhdan, D. A., Lepowsky, J., Wilson, R. L. [ 1981 ] Realization of the basic representation of the Euclidean Lie algebras, Advances in Math., 42 (1981), 83–112.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Lepowsky, J., Wilson, R. L. [ 1978 ] Construction of the affine Lie algebra All) Comm. Math. Phys. 62 (1978), 43–53.MathSciNetzbMATHGoogle Scholar
  4. Frenkel, I. B., Kac, V. G. [ 1980 ] Basic representations of affine Lie algebras and dual resonance models, Invent. Math., 62 (1980), 23–66.MathSciNetzbMATHGoogle Scholar
  5. Segal, G. [ 1981 ] Unitary representations of some infinite dimensional groups, Comm. Math. Phys. 80 (1981), 301–342.zbMATHGoogle Scholar
  6. Kac, V. G., Peterson, D. H. [ 1981 ] Spin and wedge representations of infinite dimensional Lie algebras and groups, Proc. Nat’l. Acad. Sci. USA 78 (1981), 3308–3312.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Frenkel, I. B. [ 1981 ] Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Funct. Analysis 44, (1981), 259–327.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Kostant, B. [ 1959 ] The principal three-dimensional sybgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Lepowsky, J., Wilson, R. L. [1981] A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Nat’l. Acad. Sci. USA (1981), 7254–7258.Google Scholar
  10. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. [ 1981 ] Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations III, Journal Phys. Soc. Japan, 50 (1981), 3806–3812.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. [ 1982 A] A new hierarchy of soliton equations of KP-type. Transformation groups for soliton equations IV, Physics 4D (1982), 343–365.MathSciNetzbMATHGoogle Scholar
  12. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. [ 1982 B] Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy, Publ. RIMS, Kyoto University, 18 (1982), 1077–1110.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. [1983 A] Transformation groups for soliton equations, in Proceedings of RIMS symposium (ed. M. Jimbo, T. Miwa ), 39–120, World Scientific, 1983.Google Scholar
  14. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. [ 1983 B] Landau-Lifshitz equation: solitons, quasi-periodic solutions and infinite-dimensional Lie algebras, J. Phys. A.: Math. gen. 16 (1983), 221–236.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Frenkel, I. B. [ 1982 ] Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equtions, Lecture Notes in Math. 933 (1982), 71–110.MathSciNetCrossRefGoogle Scholar
  16. Frenkel, I. B., Kac, V. G. [ 1980 ] Basic representations of affine Lie algebras and dual resonance models, Invent. Math., 62 (1980), 23–66.MathSciNetzbMATHGoogle Scholar
  17. Mandelstam, S. [ 1974 ] Dual resonance models, Physics Rep. 13 (1974), 259–353.CrossRefGoogle Scholar
  18. Schwartz, J. H. [ 1973 ] Dual resonance theory, Physics Rep. 8 (1973), 269–335.CrossRefGoogle Scholar
  19. Frenkel, I. B. [ 1983 ] Representations of Kac-Moody algebras and dual resonance models IAS, preprint.Google Scholar
  20. Kac, V. G., Peterson, D. H. [ 1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprint.Google Scholar
  21. Sato, M. [ 1981 ] Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyroku 439 (1981) 30–40.Google Scholar
  22. Garland, H., Raghunathan, M. S. [ 1975 ] A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott, Proc. Nat’l. Acad. Sci. USA 72 (1975), 4716–4717.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Pressley, A. N. [ 1980 ] Decompositions of the space of loops on a Lie group, Topology 19 (1980), 65–79.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Kazhdan, D. A., Lusztig, G. [ 1979 ] Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.MathSciNetzbMATHGoogle Scholar
  25. Tits, J. [ 1981 ] Resumé de cours, Annuaire du Collège de France 1980–81, Collège de France, Paris.Google Scholar
  26. Tits, J. [ 1982 ] Resumé de cours, Annuaire du Collège de France 1981–82, Collège de France, Paris.Google Scholar
  27. Rocha, A., Wallach, N. R. [ 1983 B] Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas, Trans. Amer. Math. Soc., 277 (1983), 133–162.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Peterson, D. H., Kac, V. G. [ 1983 ] Infinite flag varieties and conjugacy theorems, Proc. Nat’l. Acad. Sci. USA, 80 (1983), 1778–1782.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Goodman, R., Wallach, N. [ 1983 ] Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, preprint.Google Scholar
  30. Adler, M., Moser, J. [ 1978 ] On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), 1–30.MathSciNetzbMATHGoogle Scholar
  31. Vilenkin, N. Ja. [ 1965 ] Special functions and the theory of group representations, Nauka, Moscow, 1965. English translation: Providence, AMS, 1968.Google Scholar
  32. Segal, G., Wilson, G. [ 1983 ] Loop groups and equations of KdV type, Oxford University, preprint.Google Scholar
  33. Cherednik, I. B. [1983] On the definition of r-functions for generalized affine Lie algebras, Funkt. analis i ego prilozh. 17 (1983) No. 3, 93–95 (in Rusiian).MathSciNetGoogle Scholar
  34. Adler, M., van Moerbeke, P. [ 1980 A] Completely integrable systems, Euclidean Lie algebras and curves, Advances in Math. 38 (1980), 267–317.zbMATHCrossRefGoogle Scholar
  35. Adler, M., van Moerbeke, P. [ 1980 B] Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Advances in Math. 38 (1980), 318–379.zbMATHCrossRefGoogle Scholar
  36. Reiman, A. G., Semenov-Tjan-Shanskii, M. A. [ 1979 ] Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math. 54 (1979), 81–100.Google Scholar
  37. Reiman, A. G., Semenov-Tjan-Shanskii, M. A. [ 1981 ] Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II, Invent. Math. 63 (1981), 423–432.Google Scholar
  38. Ueno, K., Takasaki, K. [ 1983 ] Toda lattice heirarchy, Kyoto University, preprint.Google Scholar
  39. Drinfeld, V. G., Sokolov, V. V. [1981] Equations of Korteweg-de Vries type and simple Lie algebras, Doklady AN SSSR 258 (1981), No. 1, 11–16.MathSciNetGoogle Scholar
  40. Drinfeld, V. G., Sokolov, V. V. [ 1981 ] Equations of Korteweg-de Vries type and simple Lie algebras, English translation: Soviet Math. Doklady 23 (1981), 457–462.Google Scholar
  41. Drinfeld, V. G., Sokolov, V. V. [ 1983 ] Lie algebras and equations of KdV type, preprint (in Russian). Duflo, M. [1982] Letter to the author, dated September 19, 1982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Victor G. Kac
    • 1
  1. 1.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

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