Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 044, 29 pages      arXiv:1701.07279      https://doi.org/10.3842/SIGMA.2017.044
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Integrable Structure of Multispecies Zero Range Process

Atsuo Kuniba a, Masato Okado b and Satoshi Watanabe a
a) Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b) Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Received January 26, 2017, in final form June 07, 2017; Published online June 17, 2017

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.

Key words: integrable zero range process; stochastic $R$ matrix; matrix product formula.

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