Integrable Structure of Multispecies Zero Range Process

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.


Introduction
Zero range processes (ZRPs) [44] model a variety of stochastic dynamics in biology, chemistry, networks, physics, traffic flows and so forth. Their rich behaviors like condensation, current fluctuations and hydrodynamic limit have been important issues in non-equilibrium physics. 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. The key ingredients in these results are the stochastic R matrix and the Zamolodchikov-Faddeev (ZF) algebra. The subject lies in the intersection of quantum integrable systems and non-equilibrium statistical mechanics. As the title of the paper suggests, we will mainly focus on the former aspect, although we believe the results are essential for analyzing the physics of the model as far as the stationary properties are concerned.
Quantum R matrices are solutions of the Yang-Baxter equation (YBE) [3] and play a most fundamental role in quantum integrable systems [24]. They can be systematically produced from the representation theory of quantum groups. It remains, however, a nontrivial problem if an R matrix can be made stochastic, namely whether it can be modified so as to match the basic criteria of Markov matrices which are non-negativity and total probability conservation.
Our stochastic R matrices [29] fulfill the criteria. They originate in the quantum R matrix of the Drinfeld-Jimbo quantum affine algebra U q A (1) n in the symmetric tensor representation This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html of general degree. Plainly, they are of type A with arbitrary rank and spin, covering many examples that had been known earlier. Being higher in rank and being analytically continued in spin, it leads to systems with many kinds of particles allowing arbitrarily multiple occupancy at each lattice site, which are characteristic to multispecies ZRPs. These features are reviewed in Sections 2 and 3 based on [29]. Sections 2.3 and 3.2 also include ZRPs with a new mixed boundary condition.
It contains the stochastic R matrixŠ(λ, µ) as the structure function. Here λ, µ can be understood as generic parameters as long as algebraic aspects are concerned, but they are restricted to real numbers in a certain range in the application to the ZRP. The ZF algebra, originally introduced in the factorized scattering theories in (1+1) dimension [18,51], has penetrated into the matrix product method in integrable Markov processes in various guises since the 90's. See general remarks in Section 5.1 and also [1,6,10,14,41].
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 X α 1 ,α 2 (µ) = µ −α 1 −α 2 (µ; q) α 1 +α 2 (q; q) α 1 (q; q) α 2 (1 − zq i ) is the q-shifted factorial. For general n, the matrix product operator X α (µ) acts on the tensor product of 1 2 n(n − 1) Fock spaces. There are numerous matrix product formulas in terms of bosons known in the literature, most typically for the exclusion processes. See [1,6,14,15,30,40] for example and references therein. Our result (Theorem 5.2) is the first example distinct from them involving a quantum dilogarithm type infinite product of q-bosons.
One of the key facts in our approach is the explicit factorized formula (2.8) of an R matrix of U q A (1) n at a special point of the spectral parameter. In the last Section 6 we seek a similar result for a family of R matrices associated with the generalized quantum groups labeled by ( 1 , . . . , n+1 ) ∈ {0, 1} n+1 . They are constructed from (n + 1)-fold product of the solutions to the tetrahedron equation [50], a three-dimensional (3D) generalization of the YBE, called 3D R ( i = 0) and 3D L ( i = 1) [36]. The stochastic R matrix in Sections 2-5 originates in the special case 1 = · · · = n+1 = 0. We present the Serre type relations of the relevant generalized quantum groups explicitly and prove a similar factorized formula for ( 1 , . . . , n+1 ) of the form (1 κ , 0 n+1−κ ) (0 ≤ κ ≤ n + 1). These results are new. Their application is yet to be explored.
The layout of the paper is as follows. In Section 2 we recall two kinds of stochastic R matrices S(z) and S(λ, µ), and construct several kinds of commuting transfer matrices from them. In Section 3 we specialize these transfer matrices to formulate integrable multispecies ZRPs. They include discrete and continuous time models with both periodic and mixed boundary conditions. The latter is new. In Section 4 stationary states of these ZRPs are studied, and its matrix product construction is linked to the ZF algebra for the models with periodic boundary condition. In Section 5 we make general remarks on ZF algebra and give a q-boson representation when the structure function is the stochastic R matrix S(λ, µ). It yields the stationary probabilities in the matrix product form for the associated n-species ZRP. This part is a review of [34,35]. In Section 6 we extend the factorization (2.8) to the R matrices for a class of generalized quantum groups. The result is presented with some background connected to the tetrahedron equation [36]. Section 7 is a short summary. Appendix A contains the explicit form of the quantum R matrix for the generalized quantum group U A (1, 1, 0).
Throughout the paper we use the notation Z n = Z/nZ, θ(true) = 1, θ(false) = 0, the q-shifted The symbols (z) m appearing in this paper always mean (z; q) m . For integer arrays α = (α 1 , . . . , α m ), β = (β 1 , . . . , β m ) of any length m, we write |α| = α 1 + · · · + α m and the Kronecker The letter δ will also be used extensively to mean a local state and in such circumstances we will use the notation θ(α = β) more frequently than δ α,β to avoid confusion. The relation α ≤ β or equivalently β ≥ α is defined by β − α ∈ Z m ≥0 . We often denote by 0 or 0 m to mean (0, . . . , 0) ∈ Z m ≥0 . While preparing the text, we were informed of the paper [27], where the author obtains Markov duality functions for the models treated in this paper.

Commuting transfer matrices 2.1 Stochastic R matrices
Let us recall the stochastic R matrices S(z) and S(λ, µ) [29] associated with the Drinfeld-Jimbo quantum affine algebra U q A (1) n . They are constructed by suitably modifying the quantum R matrix characterized by (6.12).
For l ∈ Z ≥1 , introduce the vector space V l whose basis is labeled with the set B l as We write |α 1 , . . . , α n+1 simply as |α . There is an algebra homomorphism U q A (1) n → End(V l ) called the symmetric tensor representation of degree l depending on a spectral parameter. We are concerned with the standard quantum R matrix R(z) = R l,m (z) living in End(V l ⊗ V m ). Leaving the representation theoretical background to Section 6, we present an explicit formula: The integral encircles the origin u = 0 anti-clockwise to pick the residues. z is called the spectral parameter. Explicit formulas for R a,b,c i,j,k are available for example in [36, equation (2.2)]. The fact that R a,b,c i,j,k ∈ Z[q] can be seen from them. Owing to the factor δ a+b i+j δ b+c j+k in (2.4), the sum (2.3) consists of finitely many terms and R(z) γ,δ α,β is a rational function of z and q. The prefactor in (2.3) has been chosen so as to achieve the normalization (6.15), which will ultimately lead to (2.7) related to the total probability conservation.
The set of q-polynomials {R a,b,c i,j,k } form a solution of the tetrahedron equation [50] having an origin in the quantized coordinate ring of SL 3 [25]. It was stated that the composition (2.3) yields the quantum R matrix in [4] and proved in [33,Appendix B]. The formula (2.4) is due to [43]. See Section 6 and [33,36] for a further explanation and generalization. For recent progress on evaluating the sum (2.3), we refer to [9].
The first stochastic R matrix S(z) = S l,m (z) ∈ End(V l ⊗ V m ) is obtained just by taking the stochastic gauge of R(z) as follows: Example 2.1. Consider the simplest example S(z) = S 1,1 (z). We denote S(z) e i ,e j e k ,e l simply by S(z) i,j k,l , where e i is the i th basis vector defined in (6.13). By the graphical representation (2.14), nonzero elements are given by where 1 ≤ i = j ≤ n + 1. They lead to the n-species symmetric simple exclusion process at q = 1 and asymmetric simple exclusion process for q = 1. See Section 3.3.

Theorem 2.2 ([29]
). Set z i,j = z i /z j . Then the following equalities are valid: sum-to-unity :
Here S k,l 1,2 (z 1,2 ) for instance denotes the matrix that acts as S k,l (z 1,2 ) on the first and the second components from the left in V k ⊗ V l ⊗ V m . The S l,m (z) γ,δ α,β is an element of the matrix S l,m (z).
In (2.6) and (2.7), there is no constraint like l ≤ m in (2.8). The Φ q (γ|β; λ, µ) appearing in (2.8) is the function of n-component arrays β, γ and parameters q, λ, µ defined by By the definition Φ q (γ|β; λ, µ) = 0 unless γ ≤ β. The modification by the factor q η in (2.5) does not spoil the YBE. The point is that it can be so chosen that the sum-to-unity property (2.7) holds. It will eventually lead to the total probability conservation in the relevant stochastic models in what follows. The factorization (2.8) at the special point z = q l−m is also nontrivial, and assures the non-negativity of the transition rate manifestly in an appropriate range of parameters. We will generalize a formula like (2.8) to a wider class of R matrices in Section 6. The second stochastic R matrix S(λ, µ) is extracted essentially from (2.8)| q 2 →q by regarding q −l , q −m as parameters λ, µ. It is a linear operator on W ⊗ W with W defined by W = (α 1 ,...,αn)∈Z n ≥0 C|α 1 , . . . , α n . The basis |α 1 , . . . , α n here is labeled with an n-component array as opposed to (2.1), but we also denote it by the same symbol |α for simplicity. Then S(λ, µ) is defined by where Φ q (γ|β; λ, µ) is given by (2.9). We refer to the property S(λ, µ) γ,δ α,β = 0 unless α+β = γ +δ as weight conservation. The sum (2.10) is finite due to the weight conservation. In fact, the direct sum decomposition W ⊗ W = κ∈Z n ≥0 α+β=κ C|α ⊗ |β holds and S(λ, µ) splits into the corresponding submatrices. Note that the "difference property" commonly known for the original quantum R matrix and also S(z) in (2.5) has been lost and S(λ, µ) = S(cλ, cµ) does not hold.

Commuting transfer matrices with mixed boundary condition
Let us present a simple example of commuting transfer matrices having mixed boundary conditions. Let γ| with γ ∈ B l be the basis of the dual of V l such that γ|γ = δ γ,γ . Definẽ where γ| and |le i are regarded as sitting at the 0-th tensor component. The vector |le i is defined by (6.13). The matrix element is given byT In the diagram (2.18) it corresponds to the fixed boundary condition γ 0 = le i on the left and the free boundary condition γ L = γ on the right. Schematically we have where • is attached to emphasize the sum 2 . It forms a commuting family: This is most easily seen graphically as follows: The top left diagram depicts the productT i, l, z| m 1 ,...,m L and so does the bottom right (with opposite order). The first equality is by the normalization R l ,l (z /z)(|l e i ⊗ |le i ) = |l e i ⊗|le i (6.15), the second is by the YBE (2.6) and the last is due to the sum-to-unity property (2.7). The analogous transfer matrix can also be constructed from S(λ, µ) by modifying (2.20) as Here |0 = |0, . . . , 0 ∈ W and γ| is the basis of the dual of W such that γ|γ = δ γ,γ . We have the commuting family T (λ|µ 1 , . . . , µ L ),T(λ |µ 1 , . . . , µ L ) = 0 by the same token owing to the normalization S(λ, µ)(|0 ⊗ |0 ) = |0 ⊗ |0 , the YBE (2.12) and the sum-to-unity (2.13).
3 Integrable multispecies zero range process

Discrete time Markov process
In the previous section we have introduced four kinds of commuting transfer matrices. Here we design their specializations to be called Markov transfer matrices that give rise to discrete time Markov processes. Denoting the time variable by t we consider the four systems endowed with the following time evolutions:
Thus we have constructed, in the regimes specified in Proposition 3.1, commuting families of discrete time Markov processes labeled with l in (3.1), (3.2) (for each i) and with λ in (3.3), (3.4). For an interpretation in terms of stochastic dynamics of multispecies particles, see [29,Section 3.2]. In the systems (3.2) and (3.4), the weight, i.e., the number of particles, is not conserved.
Remark 3.2. The sum-to-unity (2.7), (2.13) of the stochastic R matrices alone does not necessarily imply the corresponding property (ii) for the transfer matrices if one stays in the periodic boundary condition. The latter can be established by also using the independence of the matrix elements of the NW indices in (2.14), i.e., α, δ in (2.8) and (2.11) except the weight conservation factor δ γ+δ α+β . See the explanation after equation (39) in [29]. Thus the specialization z = q l−m in (2.8) achieves the double benefit; the factorization manifesting the non-negativity and the 'NW-freeness' making the sum-to-unity of R matrices propagate to the transfer matrices.
, one sees that the auxiliary space 'merges' into the quantum space, therefore the commuting time evolutions T 1 , . . . , T L can apparently be described without the former space as illustrated by the following diagrams 3 for L = 3: If each arrow is formally viewed as a particle, they move around periodically to come back to the original position thereby 'stirring' themselves. Such a particle system appeared first in Yang's analysis on the 1D delta-function interaction gas [49, equation (14)]. In our setting, T p with p such that m p = min{m 1 , . . . , m L } fulfills the conditions (i), (ii) in the above, which may therefore be recognized as stochastic Yang's system. The system (3.3) also contains the similar Yang's system at λ = µ i due to the right relation in (2.15).
The Markov processes (3.11) with H = H r , H l are naturally viewed as the stochastic dynamics of n-species of particles on a ring of length L. The base vector |α 1 , . . . , α L with ≥0 represents a configuration in which there are α i,a particles of species a at the i-th site. There is no constraint on the number of particles that occupy a site. The matrices H r and H l describe the stochastic hopping of them to the right and the left nearest neighbor sites, respectively. The transition rate can be read off the first terms on the r.h.s. of (3.8) and (3.9), where the array γ = (γ 1 , . . . , γ n ) specifies the numbers of particles that are jumping out. In case of the superposition H(a, b, , q, µ) mentioned in the above, we have a mixture of such right and left movers. Note that the rate is determined from the occupancy of the departure site only and independent of that at the destination site, justifying the name "zero range process". Here is a snapshot of the system for the n = 2 case 5 : The process (3.11) with H =H is similarly interpreted as a stochastic particle system defined on a length L segment with boundaries. It consists of right movers only which will eventually exit from the right boundary: The integrable Markov processes constructed here and Section 3.1 cover several models studied earlier. When = 1, µ → 0 in H r , the nontrivial local transitions in (3.8) are limited to the case |γ| = 1. So if γ a = 1 and the other components of γ are 0, the rate reduces to q α 1 +···+α a−1 1−q αa 1−q . This reproduces the n-species q-boson process in [46] whose n = 1 case further goes back to [42]. For n = 1, there are extensive list of works including [5,7,8,12,13,22,39,45] for example. One can overview their interrelation in [27, Figs. 1 and 2]. When = 1, (µ, q) → (0, 0) in H l , a kinematic constraint ϕ(γ, β − γ) = 1≤i<j≤n γ i (β j − γ j ) = 0 occurs in (3.9). In 5 The arrangement of particles within each site does not matter.

Models associated with S(z)
Let us remark on the models associated with the stochastic R matrix S(z) = S l,m (z) (2.5). We refer to [27] for a further account.
To   .7), H(m) enjoys the sum-to-zero property (ii) mentioned after (3.11). However the non-negativity (i) is not satisfied just by adjusting the overall sign in general for m ≥ 2 6 . This is one of the difficulties that has been overcome by switching from S m,m (z) to S(λ, µ). The exception is m = 1, where one sees from Example 2.1 that H(1) (with the + sign in the above) is nothing but the Markov matrix of the n-species asymmetric simple exclusion process (ASEP) 7 . The transition rate r i,j of |i, j → |j, i satisfies r i,j : r j,i = 1 : q 2 for 1 ≤ i < j ≤ n + 1. Many results have been obtained for the n-species ASEP with general n including, for example, matrix product stationary states [40], spectral duality with respect to the Hasse diagram of sectors [2], solutions to stochastic initial value problem on infinite lattice [47], connection to the tetrahedron equation at q = 0 [30], application to generalized Macdonald polynomials [20] and so on.

Stationary states 4.1 Stationary probability
Here we consider the systems (3.1) and (3.3). Introduce the finite-dimensional subspaces of V m 1 ⊗ · · · ⊗ V m L and W ⊗L with fixed weight: where the labeling sets of the bases are given by where k ∈ Z n+1 ≥0 in (4.1) and k ∈ Z n ≥0 in (4.2). Note that B(k) = ∅ if |k| > m 1 + · · · + m L . Denote the discrete time evolutions (3.1) and (3.3) simply by |P (t + 1) = T |P (t) . They decompose into the equations on the subspaces V (k) for (3.1) and W (k) for (3.3), which we call sectors. If some components of k ∈ Z n ≥0 are 0, the system in such a sector becomes equivalent to the one with smaller n by an appropriate relabeling of the species. In view of this, we shall concentrate with no loss of generality on the situation k ∈ Z n ≥1 which we call basic sectors.
In what follows we shall abuse the terminology also to mean the unnormalized probabilities and states. We will treat the discrete time processes only since they cover the continuous time case. Thanks to the commutativity of the Markov transfer matrices, the stationary states are independent of l for (3.1) and of λ for (3.3).
Before closing the subsection, we include comments on the systems (3.4) and (3.11) with H =H, where the number of particles is not preserved. Starting from any initial condition, any state tends to the trivial one |0 n ⊗ · · · ⊗ |0 n , although the relaxation to it remains as an important problem. The clue to investigating it is the Bethe eigenvalues ofH. To construct them is a feasible task as done in [29,Section 4] for the periodic case H r and H l . We leave it for a future study.

Some examples
From now on we focus on the discrete time process defined by (3.3). The stationary state |P in the sector W (k) is characterized by We will refer to a sector W (k) also by k = (k 1 , . . . , k n ) ∈ Z n ≥0 for simplicity. It was known in the single species case n = 1 that the stationary state possesses the product measure (cf. [17,39] at least for the homogeneous case ∀ µ i = µ): where g σ i (µ i ) is the n = 1 case of the function Such a factorization, however, is no longer valid in the multispecies case n ≥ 2 making the system nontrivial and interesting even without an introduction (cf. [15]) of a reservoir. Particles of a given species must behave under the influence of the other species acting as a nontrivial dynamical background.
Proposition 4.3. Suppose the operators X α (µ) (α ∈ Z n ≥0 ) obey the relation Suppose further that dim Ker(T(µ i |µ 1 , . . . , µ L ) − 1) = 1 for some i. Then the matrix product formula (4.4) of the stationary probability holds for the system (3.3) provided that the trace is convergent and not identically zero.

.1 General remarks
Let us write (4.5) symbolically as where X(µ) = (X α (µ)) denotes a collection of operators. The quadratic relation of this form is called Zamolodchikov-Faddeev (ZF) algebra. Its associativity is guaranteed by the YBE satisfied by the structure functionŠ(λ, µ) 9 . Before presenting the specific results to our ZRP in the next subsection, we review its background briefly in this subsection. Similar contents can also be found for example in [10,14,20,41] and references therein. ZF algebra was originally introduced in the integrable quantum field theory in (1 + 1) dimension to encode the factorized scattering of particles [18,51]. The structure function therein should be a properly normalized scattering matrix satisfying unitarity to guarantee the total probability conservation in the quantum field theoretical setting. In the realm of integrable Markov processes, the situation is parallel. The ZF algebra serves as a local version of the stationary condition in the matrix product construction of the stationary states. The structure function, stochastic R matrix, should fulfill the sum-to-unity property. It was demonstrated in the proof of Proposition 4.3 how the ZF algebra leads to the stationary condition of the system in the case of a discrete time Markov process. Historically, however, such quadratic relations were utilized earlier in continuous time models as 'cancellation mechanism' or 'hat relations' [15]. In the present set-up it reads with h = ∂ ∂λŠ (λ, µ)| λ=µ . This is the derivative of (5.1) at λ = µ with the additional conditioň S(µ, µ) = id which matches (2.15). In terms of components, it reads h γ,δ α,β |γ ⊗ |δ . The relation (5.2) manifestly tells that the matrix product states Tr(X(µ) ⊗ · · · ⊗ X(µ)) are null vectors of the operator H = i∈Z L h i,i+1 in the periodic setting. In retrospect, one may compare the relation between (5.2) and (5.1) with that between the XXZ model and the six-vertex model in the light of Baxter's formula (3.5). Back to the ZF algebra itself, it is naturally embedded into the so-called RLL = LLR relation for an L operator L(λ) if there is a special index, say 0, such that S(λ, µ) β,α 0,0 = S(λ, µ) 0,0 β,α = θ(α = β = 0). In fact the matrix element of (5.3) for |0 ⊗ |0 → |α ⊗ |β gives (5.1) by the identification L(µ) α,0 = X α (µ). In view of this, construction of stationary states is elevated and embedded into that of representations of the stochastic RLL = LLR relation in which the relevant components of Tr(L(µ 1 ) ⊗ · · · ⊗ L(µ L )) are convergent and not identically zero. When the structure function is the R matrix R 1,1 of the vector representation, a universal L has been provided in [23]. Starting from it, one can cope with RLL = LLR with higher R l,m by the corresponding fusion of L's in principle [28]. Modifying it so as to fit the stochastic S l,m should be feasible by an appropriate twist (cf. [20]). In this sense there is a standard route to achieve the matrix product construction for S l,m -based models at least conceptually if not practically. On the other hand, a further intriguing feature is expected when the structure function is S(λ, µ) due to the peculiarity of its origin (2.8)- (2.11). This is one of our motivations in Sections 5.2 and 5.3. Turning to stationary probabilities, the maneuver in the proof of Proposition 4.3 elucidates that it is a part of more general problem of finding solutions to a quantum Knizhnik-Zamolodchikov type equation [19] with appropriate subsidiary conditions. Such wider problems have not been addressed for our stochastic R matrix (2.11). Implication of the sum-to-unity property (2.13) to the ZF algebra will be explained after Remark 5.3 until the end of Section 5. In the next subsection we will be concerned with a particular representation of the ZF algebra in terms of q-bosons.

q-boson representation
Let us present a q-boson representation of the ZF algebra (4.5) 10 . Here and in the next subsection we stay in the regime 0 < q < 1. From (2.11) it has the explicit form where the omitted condition γ ∈ Z n ≥0 should always be taken for granted. We find it convenient to work also with another normalization Z α (µ) specified by where g α (µ) has been defined in (4.3). The ZF algebra for the latter takes the form due to the identity Let B be the algebra generated by 1, b, c, k obeying the relations We call it the q-boson algebra. It has a basis {b i c j | i, j ∈ Z ≥0 }. Let F = m≥0 C(q)|m be the Fock space and F * = m≥0 C(q) m| be its dual on which the q-boson operators b, c, k act as where |−1 = −1| = 0 and 1 acts as the identity. They satisfy the defining relations (5.7). The bilinear pairing of F * and F is specified as m|m = δ m,m (q) m . Then m|(X|m ) = ( m|X)|m is valid and the trace is given by Tr(X) = m≥0 m|X|m (q)m . As a vector space, the q-boson algebra B has the direct sum decomposition B = C(q)1 ⊕ B fin , where B fin = r≥1 (B r + ⊕ B r − ⊕ B r 0 ) with B r + = s≥0 C(q)k s b r , B r − = s≥0 C(q)k s c r and B r 0 = C(q)k r . The trace Tr(X) is convergent if X ∈ B fin . It vanishes unless X ∈ r≥1 B r 0 when it is evaluated by Tr(k r ) = (1 − q r ) −1 .
The right arrow is already given by (5.8). In the sequel we will focus on the left arrow and denote Z α (µ) → (· · · ) simply by Z α (µ) = (· · · ). The ZF algebra (5.6) admits a "trivial" representation Z α (ζ) = K α in terms of an operator K α satisfying K 0 = 1 and K α K β = q ϕ(α,β) K α+β [34,Proposition 7], where ϕ(α, β) is defined in (2.9). Such a K α is easily constructed, for instance as 11 However this representation does not contain a creation operator b therefore leads to vanishing trace in the matrix product formula (4.4). Our Z α (ζ) given below may be regarded as a perturbation series starting from the trivial representation in terms of creation operators. For α i ∈ Z ≥0 , define the element Z α 1 ,...,αn (ζ) ∈ B ⊗n(n−1)/2 from the n = 1 case and the recursion with respect to n as follows: where X l (ζ) = g l (ζ)Z l (ζ) as in (5.5) and α + i is defined by (5.9).

(5.16)
It is handy to describe B ⊗n(n−1)/2 by introducing the copies B i,j = 1, b i,j , c i,j , k i,j of the q-boson algebras for 1 ≤ i ≤ j < n obeying (5.7) within each B i,j and [B i,j , B i ,j ] = 0 if (i, j) = (i , j ). We take them so that Z α 1 ,...,αn (ζ) ∈ 1≤i≤j<n B i,j and (5.16) reads Define the following elements in 1≤i≤j<n B i,j (actually in a certain completion of it): A i,j = k 1,j−1 k 2,j−1 · · · k i−1,j−1 c i,j−1 b i,j , c j,j−1 = 1.
It is not hard to show that the substitution of the formulas in Theorem 5.2 and (5.5) into the matrix product formula (4.4) of P(σ 1 , . . . , σ L ) leads to the convergent trace provided that the configuration (σ 1 , . . . , σ L ) belongs to a basic sector explained in Section 4.1. Remark 5.3. As a corollary of the ZF algebra (4.5) and S(λ, µ) 0,0 γ,δ = θ(γ = δ = 0) one can derive Let us comment on the implication of the sum-to-unity property (2.13) to the ZF algebra (4.5). Using the weight conservation it implies where w α = w α 1 1 · · · w αn n . Example 5.4. The generating function in (5.18) for n = 2 and w = (w 1 , w 2 ) = (x, y) can be computed as It plays an important role to investigate the stationary states in 'grand canonical ensemble' picture (cf. [16]).

R matrices of generalized quantum group
We have seen that the factorization (2.8) led to significant consequences in previous sections. Here we generalize it to a part of 2 n+1 quantum R matrices labeled with ( 1 , . . . , n+1 ) ∈ {0, 1} n+1 . The one treated so far corresponds to the choice ( 1 , . . . , n+1 ) = (0, . . . , 0). These R matrices have been obtained from the special solutions to the tetrahedron equation by a certain reduction [36]. The underlying algebra has been identified with a generalized quantum group. We shall present these results with a brief background based on [36].

Def inition of U A
In this subsection we assume that n is a positive integer. Let ( 1 , . . . , n+1 ) be a sequence of 0 or 1. In what follows the indices i, j are understood to be elements in Z n+1 . We write i ≡ j to mean i = j in Z n+1 . For i, j = 0, 1, . . . , n ∈ Z n+1 , set otherwise.
Let U A = U A ( 1 , . . . , n+1 ) be a Q(q)-algebra generated by e i ,f i , k ±1 i (i ∈ Z n+1 ) obeying the following relations. (We use the notation [u] = (q u − q −u )/(q − q −1 ).) 2) The algebra U A with the relations (6.1) and (6.2) was introduced for n ≥ 1 in [36] as a symmetry algebra characterizing solutions to the YBE obtained by the 2D reduction procedure to be explained in Section 6.2 from the tetrahedron equation [50]. Based on the observation in [36,Section 3.3], the relations (6.3)-(6.6) were supplemented when n ≥ 2 [37] by showing that the 13 We expect that the condition k ∈ Z n ≥1 can slightly be relaxed in view of the convergence of Tr(Xα 1 (µ1) · · · Xα L (µL)) in the non-basic sector (k1, k2) with k1 = 0, k2 ≥ 1 for n = 2. See (5.14). subalgebra generated by e i , f i , k i for i = 1, . . . , n is isomorphic, up to adding simple generators, to the quantized universal enveloping super algebra of type A [48], where they correspond to a q-analogue of the Serre relations. The forthcoming construction (6.19) and all the subsequent claims are valid for n ≥ 1. U A is a Hopf algebra with coproduct ∆ given by For the counit and the antipode, see [36, equation (3.4)]. To present a representation of U A , we introduce the following vector spaces: C|α 1 , . . . , α n+1 , (6.9) C|α ⊂ W. (6.10) Note that the range of the indices α i are to be understood as Z ≥0 or {0, 1} according to i = 0 or 1, respectively. In (6.10) we have written |α 1 , · · · , α n+1 simply as |α .

Construction of R matrix
We briefly review how we constructed in [36] solutions depending on ( 1 , . . . , n+1 ) ∈ {0, 1} n+1 to the YBE from the tetrahedron equation. Define the 3D R operator R ∈ End W (0) ⊗W (0) ⊗W (0) by . It may be viewed as a six-vertex model with q-boson valued Boltzmann weights in the third component.
Assign a solid arrow to F and a dotted arrow to V , and depict the matrix elements of 3D R and 3D L as To treat R and L on an equal footing we set M (0) = R and M (1) = L so that M ( ) ∈ End W ( ) ⊗ W ( ) ⊗ F . They satisfy the following type of tetrahedron equation [4,25,36] 14 . This is an equality in End i,j,k or R i,j,k signify that they act on the i, j and k-th components of W ( ) ⊗ W ( ) ⊗ W ( ) ⊗ F ⊗ F ⊗ F , and do as the identity on the other. The tetrahedron equation (6.16) with = 1 is depicted as follows: The equation (6.16) with = 0 is expressed similarly by replacing all the dotted arrows by solid ones.
Regarding (6.16) as a one-layer relation, we extend it to the (n + 1)-layer version. Let , where a i , b i and c i (i = 1, . . . , n + 1) are just distinct labels. Repeated use of (6.16) (n + 1) times leads to This is an equality in End F , where a = (a 1 , . . . , a n+1 ) is the array of labels and W ( n+1 ) . The notation b W and c W should be understood similarly. They are just copies of W defined in (6.9). One can reduce (6.17) to the YBE by evaluating the auxiliary space F away appropriately. A natural way is to take trace of (6.17) over the auxiliary space after multiplying it with x h 4 (xy) h 5 y h 6 from the left and R −1 4,5,6 from the right 15 . It results in the YBE for the R matrix obtained as where the scalar ρ(z) is inserted to control the normalization. The trace are taken with respect to the auxiliary Fock space F = 3 F signified by 3. Pictorially the matrix element (6.20) of (6.19) is expressed as follows: Here the broken arrows designate either F or V (6.8) according to i = 0 or 1, and the winding arrow does 3 F over which the trace is taken. In short (6.19) is a matrix product construction of quantum R matrix R(z) by operators satisfying the tetrahedron equation. The formula (2.3) is just the concrete form of (6.19) for ( 1 , . . . , n+1 ) = (0, . . . , 0).
It is not known if this type of construction extends much beyond the generalized quantum group U A . See [36, Section 2.8] for the list of the known results. However, the formula (6.19) is often more efficient than the fusion procedure practically. It also reveals a hidden 3D structure in a class of R matrices [4] and has led to another application to the multispecies totally asymmetric simple exclusion and zero range processes when ∀ i = 1 and ∀ i = 0 [30,32]. Except for the two cases however, these R matrices do not satisfy the sum-to-unity in general 16 and we have not found an application to stochastic systems.