Vertex operator representations for TKK algebras

It is known that, up to isomorphism, there are only one lattice S and one non-lattice semi-lattice S′ in R2. In [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342] we obtained a vertex operator representation for the baby-TKK algebra . The goal of the present paper is to study the TKK algebra , which includes the baby-TKK algebra as a subalgebra, and its vertex operator representations. As a by-product, we also obtain two representations for the baby-TKK algebra . One of the representations recovers the construction given in [S. Tan, TKK algebras and vertex operator representations, J. Algebra 211 (1999) 298–342].