# Ding-Iohara algebras and quantum vertex algebras

## TL;DR

This paper constructs quantum vertex algebras from Ding-Iohara algebras and classifies their irreducible modules, advancing the understanding of algebraic structures in quantum algebra.

## Contribution

It introduces a new family of associative algebras related to Ding-Iohara algebras and establishes their connection to quantum vertex algebras and module classification.

## Key findings

- Construction of quantum vertex algebras from Ding-Iohara related algebras
- Introduction of the algebra family (h) and their vacuum modules
- Classification of irreducible -coordinated modules

## Abstract

In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$ 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 $\phi$-coordinated modules for these quantum vertex algebras are associated to $\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules.