Principal subspaces for the quantum affine vertex algebra in type $A_1^{(1)}$

TL;DR

This paper introduces and analyzes principal subspaces of the quantum affine vertex algebra of type A_1^{(1)}, revealing their algebraic structure and bases related to Rogers-Ramanujan identities.

Contribution

It establishes the quantum vertex algebra structure of principal subspaces and constructs their quasi-particle bases, connecting quantum and classical vertex algebra theories.

Findings

Principal subspaces have quantum vertex algebra structure.

Constructed topological quasi-particle bases.

Linked bases to Rogers-Ramanujan-type identities.

Abstract

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type identities.