Quantum groups, Verma modules and $q$-oscillators: General linear case

Abstract

The Verma modules over the quantum groups $\mathrm U_q(\mathfrak{gl}_{l + 1})$ for arbitrary values of $l$ are analysed. The explicit expressions for the action of the generators on the elements of the natural basis are obtained. The corresponding representations of the quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ are constructed via Jimbo's homomorphism. This allows us to find certain representations of the positive Borel subalgebras of $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ as degenerations of the shifted representations. The latter are the representations used in the construction of the so-called $Q$-operators in the theory of quantum integrable systems. The interpretation of the corresponding simple quotient modules in terms of representations of the $q$-deformed oscillator algebra is given.