Ding-Iohara algebras and quantum vertex algebras

In this paper, we associate quantum vertex algebras to a certain family of associative algebras A˜(g) which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family of associative algebras A(h). The associated quantum vertex algebras are based on the vacuum modules for A(h), whereas ϕ-coordinated modules for these quantum vertex algebras are associated to A˜(g)-modules. Furthermore, we classify their irreducible ϕ-coordinated modules.