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Affine Lie algebras and combinatorial identities

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Lie Algebras and Related Topics

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References

  1. Andrews, G. E.: Partition theorems related to the Rogers-Ramanujan identities. J. Combinatorial Theory 2 (1967), 422–430.

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, G. E.: Some new partition theorems. J. Combinatorial Theory 2 (1967), 431–436.

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews, G. E.: The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, ed. G.-C. Rota. Addison-Wesley, Reading, Mass., 1976.

    Google Scholar 

  4. Bernstein, I. N., Gelfand, I. M., Gelfand S. I.: Structure of representations generated by highest weight vectors. Funkcional Anal. i Prilozhen. 5 (1971), 1–9. English translation: Functional Anal. Appl. 5 (1971), 1–8.

    Article  MathSciNet  Google Scholar 

  5. Bernstein, I. N., Gelfand, I. M., Gelfand, S. I.: Differential operators on the base affine space and a study of g-modules, in: Lie Groups and Their Representations, ed. I. M. Gelfand. Wiley, New York, 1975, pp. 21–64.

    Google Scholar 

  6. Bressoud, D.: A generalization of the Rogers-Ramanujan identities for all moduli. J. Combinatorial Theory 27 (1979), 64–68.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bressoud, D.: Analytic and combinatorial generalizations of the Rogers-Ramanujan identities. Memoirs Amer. Math. Soc. 24 (1980), Number 227.

    Google Scholar 

    Google Scholar 

  8. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations—Euclidean Lie algebras and reduction of the KP hierarchy, R.I.M.S. preprint 362 (1981).

    Google Scholar 

  9. Date, E., Kashiwara, M., Miwa, T.: Vertex operators and τ functions—transformation groups for soliton equations II, R.I.M.S. preprint 357 (1981).

    Google Scholar 

  10. Dyson, F. J.: Missed opportunities. Bull. Amer. Math. Soc. 78 (1972), 635–653.

    Article  MathSciNet  MATH  Google Scholar 

  11. Feingold, A., Lepowsky, J.: The Weyl-Kac character formula and power series identities. Advances in Math. 29 (1978), 271–309.

    Article  MathSciNet  MATH  Google Scholar 

  12. Frenkel, I. B.: Spinor representations of affine Lie algebras, Proc. Nat. Acad. Sci. U.S.A. 77 (1980), 6303–6306.

    Article  MathSciNet  MATH  Google Scholar 

  13. Frenkel, I. B.: Two constructions of affine Lie algebra representations and the boson-fermion correspondence in quantum field theory. J. Functional Analysis (1981).

    Google Scholar 

  14. Frenkel, I. B.: Representations of affine Lie algebras, Hecke modular forms and Korteweg-deVries type equations. These Proceedings.

    Google Scholar 

  15. Frenkel, I. B., Kac, V. G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62 (1980), 23–66.

    Article  MathSciNet  MATH  Google Scholar 

  16. Garland, H.: Dedekind’s η-function and the cohomology of infinite dimensional Lie algebras. Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 2493–2495.

    Article  MathSciNet  MATH  Google Scholar 

  17. Garland, H.: The arithmetic theory of loop algebras. J. Algebra 53 (1978), 480–551.

    Article  MathSciNet  MATH  Google Scholar 

  18. Garland, H., Lepowsky, J.: Lie algebra homology and the Macdonald-Kac formulas. Invent. Math. 34 (1976) 37–76.

    Article  MathSciNet  MATH  Google Scholar 

  19. Gordon, B.: A combinatorial generalization of the Rogers-Ramanujan identities. Amer. J. Math. 83 (1961), 393–399.

    Article  MathSciNet  MATH  Google Scholar 

  20. Jacobson, N.: Lie algebras. Wiley-Interscience, New York, 1962.

    MATH  Google Scholar 

  21. Kac, V. G.: Simple irreducible graded Lie algebras of finite growth. Izv. Akad. Nauk SSSR 32 (1968) 1323–1367. English translation: Math. USSR Izv. 2 (1968), 1271–1311.

    MathSciNet  Google Scholar 

  22. Kac, V. G.: Automorphisms of finite order of semisimple Lie algebras. Funkcional. Anal. i Prilozhen. 3 (1969), 94–96. English translation: Functional Anal. Appl. 3 (1969), 252–254.

    MathSciNet  MATH  Google Scholar 

  23. Kac, V. G.: Infinite-dimensional Lie algebras and Dedekind’s η-function. Funkcional. Anal. i Prilozhen. 8 (1974), 77–78. English translation: Functional Anal. Appl. 8 (1974), 68–70.

    Article  MathSciNet  Google Scholar 

  24. Kac, V. G.: Infinite-dimensional algebras, Dedekind’s η-function, classical Mőbius function and the very strange formula. Advances in Math. 30 (1978), 85–136.

    Article  MATH  Google Scholar 

  25. Kac, V. G.: An elucidation of “Infinite-dimensional algebras... and the very strange formula.” E (1)8 and the cube root of the modular invariant j. Advances in Math. 35 (1980), 264–273.

    Article  MATH  Google Scholar 

  26. Kac, V. G.: A remark on the Conway-Norton conjecture about the “Monster” simple group. Proc. Nat. Acad. Sci. U.S.A. 77 (1980), 5048–5049.

    Article  MathSciNet  MATH  Google Scholar 

  27. Kac, V. G., Kazhdan, D. A., Lepowsky, J., Wilson, R. L.: Realization of the basic representations of the Euclidean Lie algebras. Advances in Math. 42 (1981), 83–112.

    Article  MathSciNet  MATH  Google Scholar 

  28. Kac, V. G., Peterson, D.: Affine Lie algebras and Hecke modular forms. Bull. Amer. Math. Soc. (New Series) 3 (1980), 1057–1061.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kac, V. G., Peterson, D.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. U. S. A. 78 (1981), 3308–3312.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81 (1959), 973–1032.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961), 329–387.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kostant, B.: On Macdonald’s η-function formula, the Laplacian and generalized exponents. Advances in Math. 20 (1976), 179–212.

    Article  MathSciNet  MATH  Google Scholar 

  33. Lepowsky, J.: Macdonald-type identities. Advances in Math. 27 (1978), 230–234.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lepowsky, J.: Lectures on Kac-Moody Lie algebras. Université Paris VI, spring, 1978.

    Google Scholar 

  35. Lepowsky, J.: Generalized Verma modules, loop space cohomology and Macdonald-type identities. Ann. Sci. Ecole Norm. Sup. 12 (1979), 169–234.

    MathSciNet  MATH  Google Scholar 

  36. Lepowsky, J.: Application of the numerator formula to k-rowed plane partitions. Advances in Math. 35 (1980), 179–194.

    Article  MathSciNet  MATH  Google Scholar 

  37. Lepowsky, J.: Euclidean Lie algebras and the modular function j. Proc. Symp. Pure Math., Amer. Math. Soc. 37 (1980), 567–570.

    Article  MathSciNet  MATH  Google Scholar 

  38. Lepowsky, J., Mearman, A.: An E8-approach to the Leech lattice and the Conway group. J. Algebra, to appear.

    Google Scholar 

  39. Lepowsky, J., Milne, S.: Lie algebras and classical partition identities. Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 578–579.

    Article  MathSciNet  MATH  Google Scholar 

  40. Lepowsky, J., Milne, S.: Lie algebraic approaches to classical partition identities. Advances in Math. 29 (1978), 15–59.

    Article  MathSciNet  MATH  Google Scholar 

  41. Lepowsky, J., Primc, M.: manuscript in preparation.

    Google Scholar 

  42. Lepowsky, J., Wilson, R. L.: Construction of the affine Lie algebra A (1)1 . Comm. Math. Phys. 62 (1978), 43–53.

    Article  MathSciNet  MATH  Google Scholar 

  43. Lepowsky, J., Wilson, R. L.: The Rogers-Ramanujan identities: Lie theoretic interpretation and proof. Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 699–701.

    Article  MathSciNet  MATH  Google Scholar 

  44. Lepowsky, J., Wilson, R. L.: A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Advances in Math., to appear.

    Google Scholar 

  45. Lepowsky, J., Wilson, R. L.: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations. Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 7254–7258.

    Article  MathSciNet  MATH  Google Scholar 

  46. Lepowsky, J., Wilson, R. L.: manuscript in preparation.

    Google Scholar 

  47. Macdonald, I. G.: Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972), 91–143.

    Article  MathSciNet  MATH  Google Scholar 

  48. Macdonald, I. G.: unpublished manuscript.

    Google Scholar 

  49. Macdonald, I. G.: Affine Lie algebras and modular forms. Séminaire Bourbaki 33 (1981), no 577.

    Google Scholar 

    Google Scholar 

  50. Marcuson, R.: Tits’ systems in generalized nonadjoint Chevalley groups. J. Algebra 34 (1975), 84–96.

    Article  MathSciNet  MATH  Google Scholar 

  51. Moody, R. V.: A new class of Lie algebras, J. Algebra 10 (1968), 211–230.

    Article  MathSciNet  MATH  Google Scholar 

  52. Moody, R. V.: Euclidean Lie algebras. Canad. J. Math. 21 (1969), 1432–1454.

    Article  MathSciNet  MATH  Google Scholar 

  53. Moody, R. V.: Macdonald identities and Euclidean Lie algebras. Proc. Amer. Math. Soc. 48 (1975), 43–52.

    Article  MathSciNet  MATH  Google Scholar 

  54. Rocha-Caridi, A.: Resolutions of irreducible highest weight modules over infinite dimensional graded Lie algebras. These Proceedings.

    Google Scholar 

  55. Rocha-Caridi, A., Wallach, N. R.: Projective modules over graded Lie algebras I. To appear.

    Google Scholar 

  56. Segal, G.: Unitary representations of some infinite-dimensional groups. Comm. Math. Phys. 80 (1981), 301–342.

    Article  MathSciNet  MATH  Google Scholar 

  57. Serre, J.-P.: Algebres de Lie semi-simples complexes. Benjamin, New York, 1966.

    MATH  Google Scholar 

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  1. Rutgers University, 08903, New Brunswick, NJ

    J. Lepowsky

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  1. J. Lepowsky
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David Winter

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© 1982 Springer-Verlag

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Lepowsky, J. (1982). Affine Lie algebras and combinatorial identities. In: Winter, D. (eds) Lie Algebras and Related Topics. Lecture Notes in Mathematics, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093358

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  • DOI: https://doi.org/10.1007/BFb0093358

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