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

Abstract

We introduce a category ${\hat{\mathcal{O}}}_{\mathrm{osc}}$ of $q$-oscillator representations of the quantum affine algebra $U_q({\widehat{\mathfrak{gl}}}_n)$. We show that ${\hat{\mathcal{O}}}_{\mathrm{osc}}$ 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 ${\hat{\mathcal{O}}}_{\mathrm{osc}}$ 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.

42 pages, v3: Section 4.2 revised: the proof of Theorem 4.4 in v2 streamlined with newly added Lemma 4.4, together with minor corrections, v2: Abstract and Introduction revised, together with minor corrections

References in corpus (4)

• Monoidal categorification and quantum affine algebras
• Howe duality and Kostant Homology Formula for infinite-dimensional Lie superalgebras
• Higher level $q$-oscillator representations for $U_q(C_n^{(1)}),U_q(C^{(2)}(n+1))$ and $U_q(B^{(1)}(0,n))$
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$