Matrix product solutions to the reflection equation from three dimensional integrability

Abstract

We formulate a quantized reflection equation in which $q$-boson valued $L$ and $K$ matrices satisfy the reflection equation up to conjugation by a solution to the Isaev-Kulish 3D reflection equation. By forming its $n$-concatenation along the $q$-boson Fock space followed by suitable reductions, we construct families of solutions to the reflection equation in a matrix product form connected to the 3D integrability. They involve the quantum $R$ matrices of the antisymmetric tensor representations of $U_p(A_{n-1}^{(1)})$ and the spin representations of $U_p(B_n^{(1)})$, $U_p(D_n^{(1)})$ and $U_p(D_{n+1}^{(2)})$.

20 pages, minor corrections in Eq.(96)

References in corpus (1)

• Multispecies TASEP and the tetrahedron equation

Cited by in corpus (8)

• Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A_{n-1}^{(1)})$
• Boundary matrices for the higher spin six vertex model
• New solutions to the tetrahedron equation associated with quantized six-vertex models
• Quantum spin chains from Onsager algebras and reflection $K$-matrices
• Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras
• Reflection $K$ matrices associated with an Onsager coideal of $U_p(A_{n-1}^{(1)})$, $U_p(B_n^{(1)})$, $U_p(D_n^{(1)})$ and $U_p(D_{n+1}^{(2)})$
• Tetrahedron Equation and Quantum $R$ Matrices for $q$-Oscillator Representations Mixing Particles and Holes
• Matrix product solutions to the $G_2$ reflection equation