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 $U_p(A^{(1)}_{n-1})$ and the spin representations of $U_p(B^{(1)}_{n})$, $U_p(D^{(1)}_{n})$ and $U_p(D^{(2)}_{n+1})$.