Advertisement

Affine Lie algebras, theta functions and modular forms

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

Abstract

We begin this chapter with an exposition of a theory of theta functions. Using the classical transformation properties of theta functions and the theta function identity (12.7.13), we show that the string functions are modular forms and find a transformation law for these forms. Furthermore, using the theory of modular forms, we prove the “very strange” formula (12.3.7), which in turn, is used to show that the string functions, multiplied by a “standard” cusp-form, are cusp-forms. All this is applied to find explicit formulas for the weight multiplicities and characters of integrable highest weight modules.

Keywords

Holomorphic Function Modular Form Theta Function Finite Index Multiplier System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes and comments

  1. Kac, V. G., Peterson, D. H. [1983 A] Infinite dimensional Lie algebras, theta functions and modular forms, Advances in Math., 50 (1983).Google Scholar
  2. Mumford, D. [1983] Tata lectures on theta I, Birkhäuser, Boston, 1983.zbMATHGoogle Scholar
  3. Looijenga, E. [1976] Root systems and elliptic curves, Inventiones Math. 38 (1976), 17–32.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bernstein, I. N., Schvartzman, O. [1978] Chevalley theorem for complex crystallographic Coxeter groups, Funct. Anal. Appl. 12 (1978).Google Scholar
  5. Van Asch, A. [1976] Modular forms and root systems, Math. Ann. 222 (1976), 145–170.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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
  7. Feingold, A. J., Lepowsky, J. [1978] The Weyl-Kac character formula and power series identities, Adv. Math. 29 (1978), 271–309.MathSciNetzbMATHGoogle Scholar
  8. Macdonald, I. G. [1972] Affine root systems and Dedekind’s ri-function, Inventiones Math. 15 (1972), 91–143.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Feingold, A. J., Frenkel, I. B. [1983] A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2, Math. Ann. 263, (1983), 87–144.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Kac, V. G. [1980 B] An elucidation of “Infinite dimensional algebras… and the very strange formula”. E81 and the cube root of the modular invariant j, Advances in Math. 35 (1980), 264–273.zbMATHCrossRefGoogle Scholar
  11. Conway, J. H., Norton, S. P. [1979] Monstrous moonshine, Bull. London Math. Soc., 11 (1979), 308–339.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Kac, V. G. [1980 E] A remark on the Conway-Norton conjecture about the “Monster” simple group, Proc. Nat’l. Acad. Sci. USA, 77 (1980), 5048–5049.zbMATHCrossRefGoogle Scholar
  13. Griess, R. [1982] The Friendly Giant, Invent. Math, 69 (1982), 1–102.MathSciNetzbMATHGoogle Scholar
  14. Kac, V. G., Peterson, D. H. [1980] Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. 3 (1980), 1057–1061.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

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

Personalised recommendations