Affinization of $q$-oscillator representations of $U_q(\mathfrak{gl}_n)$ (2203.12862v3)

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.