# Integrable Structure of Multispecies Zero Range Process

## TL;DR

This paper reviews the integrability of multispecies zero range processes, discussing stochastic R-matrices, stationary state construction, and introduces new transfer matrices and factorizations related to quantum groups.

## Contribution

It provides a comprehensive review and introduces novel commuting transfer matrices and factorizations for multispecies zero range processes.

## Key findings

- Introduction of new commuting Markov transfer matrices with mixed boundary conditions
- Proof of factorization of R matrices related to tetrahedron equations
- Connection to quantum affine algebra and generalized quantum groups

## Abstract

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic $R$ matrices of quantum affine algebra $U_q (A^{(1)}_n)$, matrix product construction of stationary states for periodic systems, $q$-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of $R$ matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.