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We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and weprove that they can be generated by certain sets of vertex operators. As anapplication, we consider the family of graded nonlocal$\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated withthe 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 correspondingcharacter formulae.

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