Tetrahedron equation and quantum R matrices for $q$-oscillator representations of $U_q(A_{2n}^{(2)}),U_q(C_n^{(1)})$ and $U_q(D_{n+1}^{(2)})$

Abstract

The intertwiner of the quantized coordinate ring $A_q(s{l}_3)$ is known to yield a solution to the tetrahedron equation. By evaluating their $n$-fold composition with special boundary vectors we generate series of solutions to the Yang-Baxter equation. Finding their origin in conventional quantum group theory is a clue to the link between two and three dimensional integrable systems. We identify them with the quantum $R$ matrices associated with the $q$-oscillator representations of $U_q(A_{2n}^{(2)})$, $U_q(C_n^{(1)})$ and $U_q(D_{n+1}^{(2)})$.

21 pages. Reference added. Proof of irreducibility of the generic tensor product is added and some insufficient argument for it in the previous version is fixed

References in corpus (6)

• Quantum geometry of 3-dimensional lattices
• Comment on star-star relations in statistical mechanics and elliptic gamma-function identities
• An integrable 3D lattice model with positive Boltzmann weights
• A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
• Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)
• Tetrahedron equation and quantum R matrices for infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2)

Cited by in corpus (7)

• Stochastic $R$ matrix for $U_q(A_n^{(1)})$
• New solutions to the tetrahedron equation associated with quantized six-vertex models
• Higher level $q$-oscillator representations for $U_q(C_n^{(1)}),U_q(C^{(2)}(n+1))$ and $U_q(B^{(1)}(0,n))$
• Integrable Structure of Multispecies Zero Range Process
• Tetrahedron equation and Schur functions
• Tetrahedron Equation and Quantum $R$ Matrices for $q$-Oscillator Representations Mixing Particles and Holes
• Combinatorial Yang-Baxter maps arising from tetrahedron equation