The solution of the quantum -system for arbitrary boundary
We solve the quantum version of the -system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum -system and generalize it to the fully non-commutative case. We give the relation between the quantum -system and the quantum lattice Liouville equation, which is the quantized -system.
The -systems [18, 21] satisfied by the transfer matrices of the generalized Heisenberg model or the -characters of quantum affine algebras  can be considered as discrete dynamical systems with special initial conditions. More generally, the equations of these systems can be shown  to be mutations in an infinite-rank cluster algebra . As such, their solutions under general boundary conditions [8, 5] are expected to satisfy special properties such as the Laurent property and positivity.
Among discrete dynamical systems, the cluster algebras of Fomin and Zelevinsky  hold a special place. These describe the evolution of data vectors (clusters) attached to the nodes of an infinite regular tree via mutations along the edges. Mutations are defined in such a way that the following Laurent property is guaranteed: any cluster data may be expressed as a Laurent polynomial of the cluster variables at any node of the tree. It was conjectured in  and proved in several particular cases (in particular in the so-called acyclic cases [3, 15, 1], or that of clusters arising from surfaces ) that these polynomials always have non-negative integer coefficients (Laurent positivity), a property that still awaits a good general combinatorial interpretation.
Cluster algebras turn out to be quite universal, and have found applications in various fields, such as the study of non-linear recursions, the geometry of Teichmüller space, quiver representations, wall crossing formulas etc.
The relation between the recursion satisfied by the (-) characters of KR-modules of quantum affine algebras on the one hand, and cluster algebras on the other, was found in [17, 6]. Such systems are known as -sytems or -systems when they are supplemented by special boundary conditions. It is known that such equations can be interpreted as discrete integrable systems: In the case of an type algebra, the -system was identified as the discrete Hirota equation . It is also known as the tetrahedron equation in combinatorics, and arises in the context of the Littlewood-Richardson coefficients for tensor products of irreducible representations of  and domino tilings of the Aztec diamond .
Solutions to the and -systems have been constructed by various authors . Recently, a transfer matrix solution was given for the -system in the case of arbitrary boundary conditions. The latter is also known in the combinatorics literature as frieze equation . This solution was generalized to the case of in . It amounts to representing general solutions of the system as partition functions for paths on a positively weighted graph or network. The graph is determined solely by the initial conditions.
The connection to cluster algebras is as follows [17, 6]: One shows that the admissible initial data for the -systems form a subset of the clusters a cluster algebra, and that the mutations in this algebra are local transformations which are the -system equations. Thus the expression of the solutions as partition functions for positively weighted paths implies the Laurent positivity for these particular clusters.
An important question is how to quantize such evolution equations [11, 12]. The quantization in the case of cluster algebras generally was given by Berenstein and Zelevinsky . Quantum cluster algebras are non-commutative algebras where the cluster variables obey special commutation relations depending on a deformation parameter . Mutations are defined in such a way that the Laurent property is preserved, and a positivity conjecture is also expected to hold: That is, any cluster variable may be expressed as a Laurent polynomial of the variables in any other cluster seed, with coefficients in . Quantum cluster algebras were used in  to define quantum -systems for .
In the present paper, we focus on the system for , and construct its quantum version via the cluster algebra connection. We gather a few definitions in Section 2 and construct the quantum -system in Section 3. We then express the general solution in Section 4 by use of a non-commutative transfer matrix, quantizing the solution of . The main result of the paper is Theorem 4.4, which implies an interpretation of the solution as a partition function for “quantum paths” with step weights which are non-commutative Laurent monomials in the initial data, thereby proving Laurent positivity for the relevant clusters.
The -periodic solutions of the -systems satisfy -system equations, and this generalizes to the quantum case. The -system has a fully non-commutative generalization introduced by Kontsevich in the framework of wall-crossing phenomena in non-commutative Donaldson-Thomas invariant theory. The solution was given in  for this system using the method of . We revisit this system in Section 5 and formulate a fully non-commutative (as opposed to -commutative) version of the network transfer matrices used in Section 4 to solve the -system. This gives an alternative solution for the non-commutative -system.
Finally in Section 6 we give the relation between the quantum -system and the discrete quantum Liouville equation of Faddeev et al [11, 12]. This equation can also be viewed as a non-commutative -system.
Acknowledgments. We thank L. Faddev for illuminating remarks, and the organizers of the MSRI semester program on “Random Matrix Theory, Interacting Particle Systems and Integrable Systems” where this work was completed. PDF received partial support from the ANR Grant GranMa. The work of RK is supported by NSF grant DMS-0802511.
2.1. Cluster algebras and quantum cluster algebras
We use a simplified version of the definition of Fomin and Zelevinsky  of cluster algebras of geometric type with trivial coefficients.
2.1.1. Cluster algebras of geometric type
A cluster algebra is the commutative ring generated by the union of commutative variables called cluster variables. The generators are related by rational transformations called mutations determined by an exchange matrix, which governs the discrete dynamics of the system.
For the purposes of his paper, it is sufficient to consider a cluster algebra of rank , with a seed cluster consisting of cluster variables and an skew-symmetric exchange matrix . (We will also have occasion to consider cluster algebras of infinite rank (). It will be clear from our solution that when such algebras occur they are well-defined as a completion of the finite rank case.)
Clusters are pairs where is a label of a node of a complete -tree. Each node is associated with a cluster. The edges of the tree are labeled in such a way that each node is connected to exactly one edge with label where .
The clusters at nodes and connected by an edge labeled are related to each other by a mutation, which acts as a rational transformation on the component : where
A mutation also acts on the exchange matrix , such that if and are connected by an edge labeled , and
with the notation . To define a cluster algebra, it is sufficient to give the seed at one single node. This then determines the cluster variables at all other nodes via iterated mutations.
2.1.2. Quantum cluster algebras
It is interesting to consider whether there exist non-commutative generalizations of cluster algebras, which maintain some of the properties of cluster algebras. In particular, the Laurent property  and the (conjectured in general) positivity of a cluster variable at any node as an expression in terms of the cluster variables at any other node. We considered some possible candidates in  motivated by our consideration of the integrable subcluster algebras described by -systems and -systems, as well as the Kontsevich wall-crossing formula. We also considered the specialization of these non-commutative systems to the simplest type of non-commutativity, the -deformation. Such systems were first considered by Berenstein and Zelevinsky in Ref., where they defined “quantum cluster algebras”. We give here a simplified version of their definition which is sufficient for this paper.
A quantum cluster algebra is the skew field of rational functions generated by the non-commutative cluster variables where are the labels of the complete -tree as above. At a node we have the cluster where the exchange matrix is the same as for the usual cluster algebra. The cluster variables at this node -commute:
Here are the entries of an skew-symmetric matrix . Up to a scalar multiple, we can take to be the inverse of the exchange matrix . According to the definitions of  such a matrix is “compatible” with the exchange matrix .
Equivalently, we can define , where are also non-commuting variables and the exponential is taken formally. Then the commutation relations above correspond to where .
Clusters at tree nodes and connected by an edge labeled are related by a mutation. Let . Then we define to be
The exchange matrix is the same as in the commutative case.
2.2. The -system
The -systems appear in the solution of exactly solvable models in statistical mechanics, in the Bethe ansatz of generalized Heisenberg quantum spin chains based on representations of Yangians of each simple Lie algebra [18, 21]. The transfer matrices of the model satisfy a recursion relation in the highest -weight of the -modules corresponding to the auxiliary space. These relations are called -systems. In the context of representation theory, these relations are the equations satisfied by the -characters  of Kirillov-Reshetikhin modules of the Yangians, or the associated quantum affine algebra.
2.2.1. The -system associated to
These systems provide examples of discrete integrable systems which are part of a suitable cluster algebra structure . However, in the representation-theoretical context, a special initial condition is placed on the variables (corresponding to the fact that the -character of the trivial representation is 1). Here, we dispense with this special value. Moreover, we renormalize the variables so that the solutions are positive Laurent polynomials of the initial data for any initial data. This corresponds to normalizing the cluster variables of the cluster algebras so that all coefficients are trivial.
Thus, with slight abuse of notation, we call the following system the -system:
Here, we consider the set as commutative variables. Solutions of the equation are given as functions of a choice of initial variables.
2.2.2. Initial conditions
Equations (2.5) split into two independent sets of recursion relations, since the parity of is preserved by the equations. Without loss of generality, let us restrict to the relations for .
An admissible initial data set for the -system is a set
The fundamental initial data (the “staircase”) is the set
The boundary corresponding to the initial condition is the set of points in the lattice .
A solution of the -system is an expression for in terms of for each . A general solution of the -system for arbitrary boundary was given in  and generalized to the case of the algebra, , in . The solution is given in terms of a matrix representation and is interpreted as partition functions of networks.
In the present paper, we will introduce a quantum version of the -system and derive its solutions in terms of quantum networks.
2.2.3. The cluster algebra for the -system
The formulation of the -systems as sub-cluster algebras was given in . In the case of the cluster algebra is given as follows.
Let be the cluster algebra of infinite rank generated by the fundamental seed , where is given by (2.7), and the exchange matrix has entries
where the indices refer to the first index of the ’s.
Each equation in the -system corresponds to a mutation in the cluster algebra (but not vice versa). All solutions of the -system are contained in a subset of the clusters corresponding to defined in (2.6). They are obtained from via iterated cluster mutations of the form :
with , where leaves all cluster variables unchanged apart from:
in the case where all three terms on the right hand side are cluster variables in .
3. The quantum -system
3.1. Commutation relations in the initial seed
In this note, we consider the non-commuting “quantum” version of the -system. Recall that for any cluster algebra of finite rank , a quantum cluster algebra is obtained by producing a “compatible pair” of skew-symmetric integer matrices, with , where is a diagonal matrix with positive integer entries. In turn, encodes the -commutation relations between the cluster variables of the initial cluster via .
Let be an infinite, skew-symmetric matrix such that
and such that . Then
The second condition on is a choice, a reflection symmetry imposed on the matrix which determines the matrix entries (3.1) completely.
The matrix encodes the commutation relations among the elements of the fundamental cluster variable
These commutation relations are depicted graphically in Figure 1.
3.2. The quantum -system
We define the quantum -system for the variables subject to the commutation relations (3.1) to be:
As in the commuting system (2.5), this is a “three-term” recursion relation in the variable : All the variables are determined via these equations in terms of the initial data with .
Mutations are now implemented by using the relations (3.3) in the forward direction or backward direction .
Within each admissible initial data set of the quantum -system, we have the commutation relations (see Fig.2):
for all and , with mod 2.
Suppose the variables in the admissible data set satisfy (3.4) and that for some fixed value of . Let us apply (3.3) to perform a mutation with . We must check that all the commutation relations (3.4) between any for and hold. Writing , , the new cluster variable is , given by . Let , for some . Without loss of generality, let’s assume that and , . Then by the commutation relations (3.4), we have
Henceforth, must commute with , and we obtain
in agreement with (3.4). The Lemma follows. ∎
Note that and in the same cluster commute if . From the definition of admissible data sets, we see that and do not belong to the same cluster if .
Using these commutation relations, we see that the -system relation (3.3) is exactly of the form of the quantum cluster mutation (2.4), upon the renormalization of variables . We note that the subset of mutations (3.3) which we consider in the infinite rank cluster algebra makes sense, because the product on the right hand side of a mutation has only a finite number of factors (at most three).
A finite-rank quantum cluster algebra has a Laurent property , that is, cluster variables are Laurent polynomials as functions of any cluster seed, as in the case of a commutative cluster algebra. In the quantum case, the coefficients are in . It is not completely obvious that this carries over to the current case, which has infinite rank. However we will show that the solutions of the quantum -system have the Laurent property, by constructing explicit formulas for the solutions of the quantum -system in terms of any initial data . The coefficients are in , which is the analog of the positivity property of cluster algebras .
4. Quantum networks and the general solution
Here, we generalize the results of  for the network solution of the -system in terms of arbitrary admissible initial data to the non-commutative case. The solution of (3.3) in terms of any given admissible data is expressed as a quantum network partition function.
4.1. and matrices
Let be elements of . Define the matrices
These are interpreted as an elementary transfer matrix or “chip”, along a lattice with two rows, going from left to right: or is the weight of the edge connecting the dot (entry connector) in row on the left to the dot (exit connector) in row on the right in those elementary chips:
Note that we have represented the variables as attached to the faces of the chips, separated by their edges.
A quantum network is obtained by the concatenation of such chips, forming a chain where the exit connectors of each chip in the chain are identified with the entry connectors of the next chip in the chain, while face labels are well-defined. The latter condition imposes that and arguments themselves form a chain , for instance:
corresponds to the network:
The partition function of a quantum network with matrix of weights with entry connector and exit connector is . It the sum over paths from entry to exit of the product on the edges, taken in the order they are traversed.
Let be invertible elements with relations , and in , then
where is defined by the relation . This definition implies that and .
Setting the two expressions equal, we find from the element, from the one. Since commutes with , it commutes with , and the first identity implies that , i.e. , and . The Lemma follows. ∎
Let be admissible data. Then . We associate if and if . Thus for the boundary path on the lattice, is associated with a down step and is associated with an up step.
4.2. Main Theorem
Let () above a fixed admissible boundary , that is, with .
The projection of the point onto the boundary is the set of points , the portion of boundary between the lines and , with endpoints and such that with maximal and with minimal.
Figure 3 is an example of such a projection. It is a path along the boundary points from the vertex to the vertex formed by a succession of down (SE) steps and up (NE) steps . By definition such a path, if non-empty, starts with a down step and ends up with an up step.
To any path made of steps , , we associate a matrix product as follows. We define and , with the matrices and as in (4.1) and where for any point of the form we denote by . Define
This product is the weight matrix of the network made up of a concatenation of the basic network chips of the form (4.1) determined by . Let be the corresponding quantum network.
The quantum network in Equation (4.3) with weight matrix corresponds to the path
made of a succession of steps , and with a set of vertices of the form with mod 2, , with .
The main theorem of this section is the following:
Let be the projection of onto the boundary , with endpoints and . As a function of the admissible data set ,
This is proved by induction under mutations of initial data. Let be some initial data whose boundary contains the point . For such a case, we have , , and (4.7) is trivially satisfied.
Assume (4.7) holds for some boundary , let us show it also holds for the boundary with , that is, is changed for one value of , and all other values of remain unchanged.
If or , then the mutation does not affect the formula (4.7), as the boundary is modified outside of the projection of onto it, whereas and hence only depends on the boundary values within the projection.
If , five situations may occur, as sketched in Fig.4 (a-e). Let the projection of onto , and the projection of onto , and , .
If and , then . The last step of is , so . Since ,
If and , then and the last two steps of are , . Therefore,
again using .
Theorem 4.4 may be rephrased in the language of a quantum network partition function:
The quantity is the partition function of the quantum network , with weight matrix , with entry and exit connector , where is the projection of onto the boundary.
As all weights involved in the network are Laurent monomials of the initial data with coefficients in , we deduce the following positivity result, the quantum version of the Fomin-Zelevinsky positivity conjecture:
The expression for in terms of any initial data is a Laurent polynomial with coefficients in .
Consider the case where , with a boundary projection of the form: . Then , . It consists of the steps , hence
The five monomials forming are the weights of the five paths in the network with weight matrix :
Finally, we can use the symmetry of the -system under the bar involution to compute with . Given a boundary and a point above it with , let denote the projection of onto the boundary with endpoints and . Then