Universal R-matrix and functional relations

Abstract

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group $U_q(\mathcal{L}({\mathfrak{sl}}_2))$ related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

53 pages, the version submitted to the "Reviews in Mathematical Physics"

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Cited by in corpus (20)

• Matrix product solutions of boundary driven quantum chains
• Universal integrability objects
• Oscillator versus prefundamental representations II. Arbitrary higher ranks
• Oscillator versus prefundamental representations
• Quantum groups and functional relations for higher rank
• Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators
• The universal R-matrix and factorization of the L-operators related to the Baxter Q-operators
• Monodromy operators for higher rank
• Quantum groups, Verma modules and $q$-oscillators: General linear case
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Vertex Models and Spin Chains in Formulas and Pictures
• The non-compact XXZ spin chain as stochastic particle process
• Reduced qKZ equation: general case
• Quantum groups and functional relations for arbitrary rank
• $\ell$-weights and factorization of transfer operators
• Khoroshkin-Tolstoy approach for quantum superalgebras
• Highest $\ell$-Weight Representations and Functional Relations
• Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin
• Quantum affine algebras and universal functional relations
• Baxter Q- operator and functional relations