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

Abstract

The Verma modules over the quantum groups 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 are constructed via Jimbo’s homomorphism. This allows us to find certain representations of the positive Borel subalgebras of 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.