Quantum Affine Algebras and Universal $R$-Matrix with Spectral Parameter

Abstract

Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie algebras $U_q(A_1)$ and $U_q(A_2)$. As their applications, we reproduce the well-known results in the fundamental representations and we also derive an extreamly explicit formula of the spectral-dependent $R$-matrix for the adjoint representation of $U_q(A_2)$, the simplest non-trival case when the tensor product decomposition of the representation with itself has finite multiplicity.

10 pages

References in corpus (2)

• Diagonalization of the XXZ Hamiltonian by Vertex Operators
• On Universal $R$-Matrix for Quantized Nontwisted Rank 3 Affine Lie Algebras

Cited by in corpus (17)

• Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• On the Construction of Trigonometric Solutions of the Yang-Baxter Equation
• Universal R-matrix and functional relations
• Universal integrability objects
• 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
• Monodromy operators for higher rank
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Solutions of the Yang-Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras
• 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
• Universal Baxterization for $\mathbb{Z}$-graded Hopf algebras
• Quantum affine algebras and universal functional relations