Stochastic $R$ matrix for $U_q(A_n^{(1)})$

Abstract

We show that the quantum $R$ matrix for symmetric tensor representations of $U_q(A_n^{(1)})$ satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the $q$-Hahn process for $n=1$. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of $n$ species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.

21 pages. Remark 9 added, Typos in Appendix A fixed

References in corpus (6)

• Quantum geometry of 3-dimensional lattices
• Q-operators in the six-vertex model
• Self-Duality for the Two-Component Asymmetric Simple Exclusion Process
• A deformation of affine Hecke algebra and integrable stochastic particle system
• Algebraic construction of multi-species q-Boson system
• Inhomogeneous generalization of multispecies totally asymmetric zero range process

Cited by in corpus (30)

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• Non-compact quantum spin chains as integrable stochastic particle processes
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• Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz
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• Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model
• A new generalisation of Macdonald polynomials
• Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A_{n-1}^{(1)})$
• Exact Formulas of the Transition Probabilities of the Multi-Species Asymmetric Simple Exclusion Process
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• Observables of coloured stochastic vertex models and their polymer limits
• Uphill in reaction-diffusion multi-species interacting particles systems
• Boundary matrices for the higher spin six vertex model
• The non-compact XXZ spin chain as stochastic particle process
• Matrix product formula for $U_q(A_2^{(1)})$-zero range process
• Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model
• Observables of stochastic colored vertex models and local relation
• Stochastic symplectic ice
• A $q$-boson representation of Zamolodchikov-Faddeev algebra for stochastic $R$ matrix of $U_q(A_n^{(1)})$
• Integrable boundaries for the q-Hahn process
• Current statistics in the q-boson zero range process
• Integrable Structure of Multispecies Zero Range Process
• On the eigenfunctions for the multi-species q-Boson system
• Density and current profiles in $U_q(A_2^{(1)})$ zero range process
• Inhomogeneous spin $q$-Whittaker polynomials
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters
• Explicit Central Elements of $U_q(\mathfrak{gl}(N+1))$
• Triangular solutions to the reflection equation for $U_q(\widehat{s{l}_n})$