Principal Vertex Operator Representations For Toroidal Lie Algebras

Vertex operators discovered by physicists in string theory have turned out to be important objects in mathematics. One can use vertex operators to construct various realizations of the irreducible highest weight representations for affine Kac-Moody algebras. Two of these, the principal and homogeneous realizations, are of particular interest. The principal vertex operator construction for the affine algebra A (1) 1 allows one to construct soliton solutions of the Korteweg de Vries hierarchy of partial differential equations. On the other hand, the homogeneous realization is linked to the fundamental nonlinear Schrödinger equation [Kac]. S.Eswara Rao and R.V. Moody in [EM] studied the homogeneous vertex operator construction for toroidal Lie algebras. The present paper is devoted to the principal realization. Here we construct the principal vertex operator representation for the toroidal Lie algebra ĝ which is a universal central extension of g̃ = ġ ⊗ C[t±0 , . . . , t ± n ], in which ġ is a simply-laced simple finite-dimensional Lie algebra over C. This generalizes the principal vertex operator realization of the basic representations of affine Lie algebras constructed in [KKLW]. We add the Lie algebraD of vector fields on a torus to the toroidal Lie algebra ĝ to form a larger algebra g. This is necessary in order to have a sufficiently large principal Heisenberg subalgebra. To construct a representation of g we consider the standard representation of the principal Heisenberg subalgebra on the Fock space F and then extend it to all of g by