Affinization of $q$-oscillator representations of $U_q(\mathfrak{gl}_n)$

TL;DR

This paper introduces a new category of $q$-oscillator representations for quantum affine algebra $U_q(\\mathfrak{gl}_n)$, connecting finite-dimensional irreducible representations of affine type $A$ and superalgebras, enriching the understanding of their structure.

Contribution

It constructs a category of $q$-oscillator representations that interpolates between different families of irreducible representations of quantum affine algebras and superalgebras.

Findings

Established a family of irreducible representations within the category.

Connected $q$-oscillator representations to finite-dimensional irreducible representations.

Provided a quantum affine analogue of a tensor category related to $\\mathfrak{gl}_{u+v}$ duality.

Abstract

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.