A $q$-boson representation of Zamolodchikov-Faddeev algebra for stochastic $R$ matrix of $U_q(A^{(1)}_n)$

TL;DR

This paper develops a $q$-boson representation of the Zamolodchikov-Faddeev algebra linked to the stochastic $R$ matrix of $U_q(A^{(1)}_n)$, enabling a matrix product formula for stationary probabilities in a quantum zero-range process.

Contribution

It introduces a novel $q$-boson representation involving quantum dilogarithm products for the algebra associated with the stochastic $R$ matrix of $U_q(A^{(1)}_n)$.

Findings

Derived a matrix product formula for stationary probabilities.

Connected the algebraic structure to a quantum zero-range process.

Utilized infinite product representations in the $q$-boson framework.

Abstract

We construct a $q$-boson representation of the Zamolodchikov-Faddeev algebra whose structure function is given by the stochastic $R$ matrix of $U_q(A^{(1)}_n)$ introduced recently. The representation involves quantum dilogarithm type infinite products in the $n(n-1)/2$-fold tensor product of $q$-bosons. It leads to a matrix product formula of the stationary probabilities in the $U_q(A_n^{(1)})$-zero range process on a one-dimensional periodic lattice.