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

The Verma modules over the quantum groups {U}_q(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 {U}_q({ L}(sl_{l + 1})) are constructed via Jimbo’s homomorphism. This allows us to find certain representations of the positive Borel subalgebras of {{U}}_q({ L}(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.