Higher level vertex operators for $U_q (\hat{\mathfrak{sl}}_2)$

TL;DR

This paper explores graded nonlocal q-vertex algebras related to quantum affine algebra modules, establishing their generation, bases, and character formulas, advancing understanding of their algebraic structure and combinatorial properties.

Contribution

It introduces the generation of graded nonlocal q-vertex algebras by specific vertex operators and derives combinatorial bases and character formulas for modules of quantum affine algebra.

Findings

Generated graded nonlocal q-vertex algebras by specific sets of vertex operators.

Derived combinatorial bases for modules associated with quantum affine algebra.

Computed explicit character formulas for these modules.

Abstract

We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(c\Lambda_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(c\Lambda_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.