A $q$-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra

We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra $\mathrm U_q(\widehat{\mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand–Tsetlin character or ($q$-character). We also apply the quantum Sylvester theorem to construct a $q$-analogue of the Olshanski algebra as a projective limit of certain centralizers in $\mathrm U_q(\mathfrak{gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra.