Vertex operators and principal subspaces of level one for $U_q (\widehat{\mathfrak{sl}}_2)$

TL;DR

This paper explores two methods of constructing vertex algebraic structures for level 1 principal subspaces of $U_q(\\widehat{\rak{sl}}_2)$, revealing combinatorial bases and connections to Rogers-Ramanujan identities.

Contribution

It introduces commutative and nonlocal q-vertex algebra frameworks, linking algebraic bases to classical combinatorial identities and quantum quasi-particle relations.

Findings

Identified combinatorial bases matching Feigin-Stoyanovsky bases.

Established connections between nonlocal q-vertex algebras and Rogers-Ramanujan identities.

Discussed applications to quantum quasi-particle relations.

Abstract

We consider two different methods of associating vertex algebraic structures with the level $1$ principal subspaces for $U_q (\widehat{\mathfrak{sl}}_2)$. In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B. L. Feigin and A. V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal $\underline{\mathsf{q}}$-vertex algebras, investigate their properties and construct the nonlocal $\underline{\mathsf{q}}$-vertex algebra and its module, generated by Frenkel-Jing operator and Koyama's operator respectively. By finding the combinatorial bases of their suitably defined subspaces, we establish a connection with the sum sides of the Rogers-Ramanujan identities. Finally, we discuss further applications to quantum quasi-particle relations.