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

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 (1) n )-zero range process on a one-dimensional periodic lattice.

Integrable Structure of Multispecies Zero Range Process

“…We have reviewed the construction of the multispecies ZRP in [29], the matrix product formula for the stationary probability in [34,35] and the relevant quantum R matrices originating in the tetrahedron equation and the generalized quantum groups in [36]. We have also pointed out a new commuting Markov transfer matrix in Section 2.3, the associated Markov Hamiltonian (3.10), the Serre type relations (6.3)-(6.6) for the generalized quantum group U A ( 1 , .…”

“…See for example [16,21,26] and references therein. This paper is a brief summary of the integrable multispecies ZRP in one dimension introduced and studied in the recent works [29,34,35]. We formulate the ZRPs via commuting Markov transfer matrices and present a matrix product formula for stationary probabilities in the periodic boundary condition.…”

“…We will review a q-boson representation of the ZF algebra obtained in [34,35]. The simplest nontrivial case is n = 2 for which it is (1 − zq i ) is the q-shifted factorial.…”

The non-compact XXZ spin chain as stochastic particle process

“…TAZRP Taking a limit that can be related to a single totally asymmetric zero range process (TAZRP), cf. [18,21], does not seem to be straightforward at the level of the Hamiltonian density. However, as a consequence of Baxter's TQ-equation which relates the transfer matrix and the Q-operator, the generators of the two TAZRP's may arise from the logarithmic derivative of the Q-operator of the non-compact XXZ chain at two special points of the spectral parameter.…”

Density and current profiles inUq(A2(1))Kuniba1, Mangazeev2 2017Nuclear Physics B4070View full textAdd to dashboardCite

“…. , α n ) ∈ Z n ≥0 possesses a nested structure with respect to the rank n [14]. Thus the U q (A (1) n ) ZRPs form the first systematic examples of multispecies (or multi-class) ZRPs whose stationary measure on the ring is not factorized.…”