Geometriae Dedicata

, Volume 35, Issue 1, pp 283–307

Toroidal Lie algebras and vertex representations


  • Robert V. Moody
    • Department of MathematicsUniversity of Alberta
  • Senapathi Eswara Rao
    • School of MathematicsTata Institute for Fundamental Research
  • Takeo Yokonuma
    • Department of MathematicsSophia University

DOI: 10.1007/BF00147350

Cite this article as:
Moody, R.V., Rao, S.E. & Yokonuma, T. Geom Dedicata (1990) 35: 283. doi:10.1007/BF00147350


The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ℂ××ℂ× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.

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© Kluwer Academic Publishers 1990