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.

18 pages, LaTeX2e; checked for typos and minor corrections are made; version to appear in J. Phys. A: Math. Theor

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Cited by in corpus (7)

• Oscillator versus prefundamental representations II. Arbitrary higher ranks
• A note on $q$-oscillator realizations of $U_q(gl(M|N))$ for Baxter $Q$-operators
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Vertex Models and Spin Chains in Formulas and Pictures
• Quantum groups and functional relations for arbitrary rank
• $\ell$-weights and factorization of transfer operators
• Highest $\ell$-Weight Representations and Functional Relations