Principal construction of the toroidal Lie algebra of type \(A_1\)
A toroidal Lie algebra is a perfect central extension of the Lie algebra \({\mathcal G}\otimes A_\nu\) where \({\mathcal G}\) is a finite-dimensional simple Lie algebra, and \(A_\nu\) is the ring of Laurent polynomials in commuting variables. In this paper we consider an analogue of the level one principal representation of the affine Kac-Moody algebra \(A_1^{(1)}\) by vertex operators to the toroidal case. We construct a module for the toroidal Lie algebra and an extended toroidal algebra of type \(A_1\). The Fock module representation obtained here is faithful and completely reducible over the extended toroidal Lie algebra. Some structure of the module is also studied in the paper.
scientific article; zbMATH DE number 1336851