Cluster algebras and discrete integrability
CCluster algebras and discrete integrability
Andrew N.W. Hone ∗ , Philipp Lampe and Theodoros E. KouloukasSchool of Mathematics, Statistics & Actuarial ScienceUniversity of Kent, Canterbury CT2 7FS, UK.March 21, 2019 Abstract
Cluster algebras are a class of commutative algebras whose generatorsare defined by a recursive process called mutation. We give a briefintroduction to cluster algebras, and explain how discrete integrablesystems can appear in the context of cluster mutation. In particular,we give examples of birational maps that are integrable in the Liouvillesense and arise from cluster algebras with periodicity, as well as exam-ples of discrete Painlev´e equations that are derived from Y-systems.
Cluster algebras are a special class of commutative algebras that were intro-duced by Fomin and Zelevinsky almost twenty years ago [21], and rapidlybecame the hottest topic in modern algebra. Rather than being defined apriori by a given set of generators and relations, the generators of a clusteralgebra are produced recursively by iteration of a process called mutation.In certain cases, a sequence of mutations in a cluster algebra can correspondto iteration of a birational map, so that a discrete dynamical system is gen-erated. The reason why cluster algebras have attracted so much attention isthat cluster mutations and associated discrete dynamical systems or differ-ence equations arise in such a wide variety of contexts, including Teichmullertheory [19, 20], Poisson geometry [29], representation theory [11], and inte-grable models in statistical mechanics and quantum field theory [14, 33, 66],to name but a few.The purpose of this review is to give a brief introduction to cluster al-gebras, and describe certain situations where the associated dynamics is ∗ Currently on leave at UNSW, Sydney, Australia. a r X i v : . [ m a t h . C O ] M a r ompletely integrable, in the sense that a discrete version of Liouville’s the-orem in classical mechanics is valid. Furthermore, within the context ofcluster algebras, we will describe a way to detect whether a given discretesystem is integrable, based on an associated tropical dynamical system andits connection to the notion of algebraic entropy. Finally, we describe howdiscrete Painlev´e equations can arise in the context of cluster algebras. A cluster algebra with coefficients, of rank N , is generated by starting froma seed ( B, x , y ) consisting of an exchange matrix B = ( b ij ) ∈ Mat N ( Z ),an N -tuple of cluster variables x = ( x , x , . . . , x N ), and another N -tupleof coefficients y = ( y , y , . . . , y N ). The exchange matrix is assumed to beskew-symmetrizable, meaning that there is a diagonal matrix D , consistingof positive integers, such that DB is skew-symmetric. For each integer k ∈ [1 , N ], there is a mutation µ k which produces a new seed ( B (cid:48) , x (cid:48) , y (cid:48) ) = µ k ( B, x , y ). The mutation µ k consists of three parts: matrix mutation ,which is applied to B to produce B (cid:48) = ( b (cid:48) ij ) = µ k ( B ), where b (cid:48) ij = (cid:40) − b ij if i = k or j = k,b ij + sgn( b ik )[ b ik b kj ] + otherwise , (1)with sgn( a ) being ± a ∈ R and 0 for a = 0, and[ a ] + = max( a, coefficient mutation , defined by y (cid:48) = ( y (cid:48) j ) = µ k ( y ) where y (cid:48) j = y − k if j = k,y j (cid:16) y − sgn( b jk ) k (cid:17) − b jk otherwise; (2)and cluster mutation , given by x (cid:48) = ( x (cid:48) j ) = µ k ( x ) with the exchange relation x (cid:48) k = y k (cid:81) Ni =1 x [ b ki ] + i + (cid:81) Ni =1 x [ − b ki ] + i (1 + y k ) x k , (3)and x (cid:48) j = x j for j (cid:54) = k .Given an initial seed, one can apply an arbitrary sequence of mutations,which produces a sequence of seeds. This can be visualized by attaching theinitial seed to the root of an N -regular tree T N (with N branches attached2o each vertex), and then labelling the seeds as ( B t , x t , y t ) with “time” t ∈ T N . Note that mutation is an involution, µ k · µ k = id, but in generaltwo successive mutations do not commute, i.e. typically µ j · µ k (cid:54) = µ k · µ j for j (cid:54) = k . Moreover, in general the exponents and coefficients appearing in theexchange relation (3) change at each stage, because the matrix B and the y variables are altered by each of the previous mutations. Definition 1.
The cluster algebra A ( B, x , y ) is the algebra over C ( y ) gener-ated by the cluster variables produced by all possible sequences of mutationsapplied to the seed ( B, x , y ).We will also consider the case of coefficient-free cluster algebras, forwhich the y variables are absent, the seeds are just ( B, x ), and the clustermutation is defined by the simpler exchange relation x (cid:48) k = (cid:81) Ni =1 x [ b ki ] + i + (cid:81) Ni =1 x [ − b ki ] + i x k . (4) Remark 1.
The original definition of a cluster algebra in [21] involves amore general setting in which the coefficients y are elements of a semifield P , that is, an abelian multiplicative group together with a binary operation ⊕ that is commutative, associative and distributive with respect to multipli-cation. In that setting, with the N -tuple y ∈ P N , the algebra A ( B, x , y ) isdefined over Z [ P ], and the addition in the denominator of (3) is given by ⊕ .The case we consider here corresponds to P = P univ , the universal semifield,consisting of subtraction-free rational functions in the variables y j , in whichcase ⊕ becomes ordinary addition in the field of rational functions C ( y ).However, starting with the more general setting, we can also consider thecase of the trivial semifield with one element, P = { } , which yields thecoefficient-free case (4).In order to illustrate the above definitions, we now present a numberof concrete examples. For the sake of simplicity, we concentrate on thecoefficient-free case in the rest of this section, and return to the equationswith coefficients y at a later stage. Example 1. The cluster algebra of type B : A particular cluster alge-bra of rank N = 2 is given by taking the exchange matrix B = (cid:18) − (cid:19) , (5)3nd the initial cluster x = ( x , x ), to define a seed ( B, x ). The matrix B isskew-symmetrizable: the diagonal matrix D = diag(1 ,
2) is such that DB = (cid:18) − (cid:19) is skew-symmetric. Applying the mutation µ and using the rule (1) givesa new exchange matrix B (cid:48) = µ ( B ) = (cid:18) −
21 0 (cid:19) = − B, while the coefficient-free exchange relation (4) gives a new cluster x (cid:48) =( x (cid:48) , x ) with x (cid:48) = x + 1 x . Since mutation acts as an involution, we have µ ( B (cid:48) , x (cid:48) ) = ( B, x ), so noth-ing new is obtained by applying µ to this new seed. Thus we consider µ ( B (cid:48) , x (cid:48) ) = µ · µ ( B, x ) instead, which produces µ ( B (cid:48) ) = B and µ ( x (cid:48) ) =( x (cid:48) , x (cid:48) ), where x (cid:48) = x (cid:48) + 1 x = x + x + 1 x x . Once again, a repeat application of the same mutation µ returns to theprevious seed, so instead we consider applying µ to obtain µ · µ · µ ( B ) = − B and µ · µ · µ ( x ) = ( x (cid:48)(cid:48) , x (cid:48) ), with x (cid:48)(cid:48) = x + 2 x + x + 1 x x . Repeating this sequence of mutations, it is clear that the exchange matrixjust changes by an overall sign at each step. Perhaps more surprising is thefact that after obtaining ( µ · µ ) ( x ) = ( x (cid:48)(cid:48) , x (cid:48)(cid:48) ), with x (cid:48)(cid:48) = x + 1 x , the variable x reappears in the cluster after a further step, i.e. µ · ( µ · µ ) ( x ) = ( x , x (cid:48)(cid:48) ), and finally ( µ · µ ) ( x ) = ( x , x ) = x , so that theinitial seed ( B, x ) is restored after a total of six mutations. Thus the clusteralgebra has a finite number of generators in this case, since there are only thesix cluster variables x , x , x (cid:48) , x (cid:48) , x (cid:48)(cid:48) , x (cid:48)(cid:48) . This example is called the cluster4igure 1: The quiver Q corresponding to the exchange matrix (6). algebra of type B , since the initial matrix B is derived from the Cartanmatrix of the B root system, that is C = (cid:18) − − (cid:19) , by replacing the diagonal entries in C with 0, and changing signs of theoff-diagonal entries so that b ij and b ji have opposite signs for i (cid:54) = j .There are two significant features of the preceding example, namely thefact that there are only finitely many clusters, and the fact that the clustervariables are all Laurent polynomials (polynomials in x , x and their recip-rocals) with integer coefficients. The first feature is rare: a cluster algebrais said to be of finite type if there are only finitely many clusters, and itwas shown in [23] that all such cluster algebras are generated from seedscorresponding to the finite root systems that appear in the Cartan-Killingclassification of finite-dimensional semisimple Lie algebras. The second fea-ture (the Laurent phenomenon) is ubiquitous [22], and follows from thefollowing result, proved in [22]. Proposition 1.
All cluster variables in a coefficient-free cluster algebra A ( B, x ) are Laurent polynomials in the variables from the initial cluster,with integer coefficients, i.e. they are elements of the ring of Laurent poly-nomials, that is Z [ x ± ] := Z [ x ± , x ± , . . . , x ± N ] . There is an analogous statement in the case that coefficients are included,and in fact it is possible to prove the stronger result that all of the coefficientsof the cluster variables have positive integer coefficients, so they belong to Z > [ x ± ] (see [35, 52], for instance). 5 xample 2. The cluster algebra of type ˜ A , : As an example of rank N = 4, we take the skew-symmetric matrix B = − − − − , (6)which is obtained from the Cartan matrix of the affine root system A (1)3 [48],namely C = − − − − − − − − , by replacing each of the diagonal entries of C with 0, and making a suitableadjustment of signs for the off-diagonal entries, such that b ij = − b ji . Since B is a skew-symmetric integer matrix, it can be associated with a quiver Q without 1- or 2-cycles, that is, a directed graph specified by the rule that b ij is equal to the number of arrows i → j if it is non-negative, and minus thenumber of arrows j → i otherwise (see Fig.1). If the mutation µ is applied,then the new exchange matrix is B (cid:48) = µ ( B ) = − −
11 0 1 00 − − , which corresponds to a new quiver Q (cid:48) obtained by a cyclic permutationof the vertices of the original Q (see Fig.2), while the initial cluster x =( x , x , x , x ) is mutated to µ ( x ) = ( x (cid:48) , x , x , x ), where x (cid:48) is defined bythe relation x x (cid:48) = x x + 1 . Rather than trying to describe the effect of every possible choice of mutation,we consider what happens when µ is followed by µ , and once more observethat, at the level of the associated quiver, this just corresponds to applyingthe same cyclic permutation as before to the vertex labels 1,2,3,4. The newcluster obtained from this is µ · µ ( x ) = ( x (cid:48) , x (cid:48) , x , x ), with x (cid:48) defined by x x (cid:48) = x x (cid:48) + 1 , µ is applied next, then µ · µ · µ ( x ) = ( x (cid:48) , x (cid:48) , x (cid:48) , x ), with x x (cid:48) = x x (cid:48) + 1 . Continuing in this way, it is not hard to see that the composition µ · µ · µ · µ takes the original B to itself, and applying this sequence of mutationsrepeatedly in the same order generates a new cluster variable at each step,with the sequence of cluster variables satisfying the nonlinear recurrencerelation x n x n +4 = x n +1 x n +3 + 1 (7)(where we have made the identification x (cid:48) = x , x (cid:48) = x , and so on).Regardless of other possible choices of mutations, this particular sequenceof mutations alone generates an infinite set of distinct cluster variables, ascan be seen by fixing some numerical values for the initial cluster. In fact,as was noted in [42], for any orbit of (7) there is a constant K such that theiterates satisfy the linear recurrence x n +6 + x n = Kx n +3 . (8)Upon fixing ( x , x , x , x ) = (1 , , , , , , , , , , , , , , , . . . , which also satisfies the linear recurrence (8) with K = 5; so the terms growexponentially with n , and the integers x n are distinct for n ≥
4. Thisis called the ˜ A , cluster algebra, because the corresponding quiver is anorientation of the edges of an affine Dynkin diagram of type A with oneanticlockwise arrow and three clockwise arrows.The skew-symmetry of B is preserved under matrix mutation, and forany skew-symmetric integer matrix there is an equivalent operation of quivermutation which acts on the associated quiver Q : to obtain the mutatedquiver µ k ( Q ) one should (i) add pq arrows i pq −→ j whenever Q has a pathof length two passing through vertex k with p arrows i p −→ k and q arrows k q −→ j ; (ii) reverse all arrows in Q that go in/out of vertex k ; (iii) deleteany 2-cycles created in the first step.Unlike the B cluster algebra, the above example is not of finite type,because there are infinitely many clusters. However, it turns out that itis of finite mutation type, in the sense that there are only a finite numberof exchange matrices produced under mutation from the initial B . Cluster7igure 2: The mutated quiver Q (cid:48) = µ ( Q ) obtained by applying µ to (6). Figure 3:
The quiver corresponding to the exchange matrix (9). algebras of finite mutation type have also been classified [16, 17]: as wellas those of finite type, they include cluster algebras associated with trian-gulated surfaces [19, 20], cluster algebras of rank 2, plus a finite number ofexceptional cases.
Example 3. Cluster algebra related to Markoff ’s equation:
For N =3, consider the exchange matrix B = − − − , (9)which is associated with the quiver in Fig.3. After any sequence of matrixmutations, one can obtain only B or − B , so this is another example of finitemutation type: it is connected to the moduli space of once-punctured tori,8nd the Markoff equation x + y + z = 3 xyz (10)which arises in that context as well as in Diophantine approximation theory[7, 9]. Upon applying µ to the initial cluster ( x , x , x ), the result is( x (cid:48) , x , x ) with x x (cid:48) = x + x , and a subsequent application of µ yields ( x (cid:48) , x (cid:48) , x ), where x x (cid:48) = x + x (cid:48) . Repeated application of the mutations µ · µ · µ in that order produces anew cluster variable at each step, and upon identifying x = x (cid:48) , x = x (cid:48) ,and so on, the sequence of cluster variables ( x n ) is generated by a recurrenceof third order, namely x n x n +3 = x n +1 + x n +2 . (11)It can also be shown that on each orbit of (11) there is a constant K suchthat the nonlinear relation x n +3 + x n = Kx n +1 x n +2 holds for all n , and by using the latter to eliminate x n +3 it follows that K = x n + x n +1 + x n +2 x n x n +1 x n +2 (12)is an invariant for (11), independent of n . In particular, taking the ini-tial values to be (1 , ,
1) gives K = 3, and each adjacent triple ( x, y, z ) =( x n , x n +1 , x n +2 ) in the resulting sequence1 , , , , , , , , , . . . (13)is an integer solution of Markoff’s equation (10). The terms of this sequencehave double exponential growth: log x n grows exponentially with n .The next example is generic, in the sense that there are both infinitelymany clusters and infinitely many exchange matrices.9igure 4: The Somos-6 quiver corresponding to the exchange matrix (14).
Example 4. A Somos-6 recurrence:
A sequence that is generated by aquadratic recurrence relation of the form x n x n + k = (cid:98) k/ (cid:99) (cid:88) j =1 α j x n + j x n + k − j , where α j are coefficients, is called a Somos- k sequence (see [22, 28, 40, 62,63]). A certain class of Somos-6 sequences can be generated by starting fromthe exchange matrix B = − − − − − − − − − − − , (14)which corresponds to the quiver in Fig.4. Upon applying cyclic sequences ofmutations ordered as µ · µ · µ · µ · µ · µ , a sequence of cluster variables( x n ) is produced which satisfies the particular Somos-6 recurrence x n x n +6 = x n +1 x n +5 + x n +3 . (15)If six 1s are chosen as initial values, then an integer Somos-6 sequence be-ginning with 1 , , , , , , , , , , , , , , . . . is produced. For this sequence, log x n grows like n . However, applyingsuccessive mutations other than these cyclic ones generally causes the mag-10itude of the entries of the exchange matrices to grow - for instance, µ · µ · µ ( B ) = − − −
10 4 0 − − −
20 10 1 0 − − − − − ;and (e.g. starting with the initial seed evaluated as (1 , , , , ,
1) as be-fore) typically this results in the values of cluster variables showing doubleexponential growth with the number of steps.
The exchange relation (3) can be regarded as a birational map in C N . Alter-natively, x ∈ ( C ∗ ) N can be viewed as coordinates in a toric chart for somealgebraic variety, and a mutation x (cid:55)→ µ k ( x ) = x (cid:48) as a change of coordinatesto another chart. The latter point of view is passive, in the sense that thereis some fixed variety and mutation just selects different choices of coordinatecharts. Instead of this, we would like to take an active view, regarding eachmutation as an iteration in a discrete dynamical system. However, there isa problem with this, because a general sequence of mutations is specifiedby a “time” t belonging to the tree T N , and (except for the case of rank N = 2) this cannot naturally be identified with a discrete time belongingto the set of integers Z . Furthermore, there is the additional problem thatmatrix mutation, as in (1), typically changes the exponents appearing in theexchange relation, so that in general it is not possible to interpret successivemutations as iterations of the same map.Despite the above comments, it turns out that the most interesting clus-ter algebras appearing “in nature” have special symmetries, in the sense thatthey display periodic behaviour with respect to at least some subset of thepossible mutations. In fact, all of the examples in the previous section are ofthis kind. Here we consider a notion of periodicity that was introduced byFordy and Marsh [25] in the context of skew-symmetric exchange matrices B , which correspond to quivers. Definition 2.
An exchange matrix B is said to be cluster mutation-periodic with period m if (for a suitable labelling of indices) µ m · µ m − · . . . · µ ( B ) = ρ m ( B ), where ρ is the cyclic permutation ρ : (1 , , , . . . , N ) (cid:55)→ ( N, , , . . . , N − .
11n the context of quiver mutation, the case of cluster mutation-periodicitywith period m = 1 means that the action of mutation µ on Q is the sameas the action of ρ , which is such that the number of arrows i → j in Q is thesame as the number of arrows ρ − ( i ) → ρ − ( j ) in ρ ( Q ). This means that thecluster map ϕ = ρ − · µ acts as the identity on Q (or equivalently, on B ),but in general x (cid:55)→ ϕ ( x ) has a non-trivial action on the cluster. Mutation-periodicity with period 1 implies that iterating this map is equivalent toiterating a single recurrence relation. Example 5.
Although it is not skew-symmetric, the exchange matrix (5)in Example 1 is cluster mutation-periodic with period 2, since µ · µ ( B ) = ρ ( B ) = B where ρ is the switch 1 ↔
2. Defining the cluster map to be ϕ = ρ − · µ · µ , the action of ϕ is periodic with period 3 for any choice ofinitial cluster, i.e. ϕ =id. Example 6.
The exchange matrix (6) in Example 2 is cluster mutation-periodic with period 1. The cluster map ϕ = ρ − · µ is given by ϕ : ( x , x , x , x ) (cid:55)→ (cid:18) x , x , x , x x + 1 x (cid:19) , (16)whose iterates are equivalent to those of the nonlinear recurrence (7). Example 7.
The exchange matrix (9) in Example 3 is cluster mutation-periodic with period 2. The cluster map ϕ = ρ − · µ · µ is given by ϕ : ( x , x , x ) (cid:55)→ (cid:18) x , x (cid:48) , x + x (cid:48) x (cid:19) , with x (cid:48) = x + x x . (17)Each iteration of (17) is equivalent to two iterations of the nonlinear re-currence (11). This period 2 example is exceptional because, in general,cluster mutation-periodicity with period m > Example 8.
The exchange matrix (14) in Example 4 is cluster mutation-periodic with period 1. The cluster map is given by ϕ : ( x , x , x , x , x , x ) (cid:55)→ (cid:18) x , x , x , x , x , x x + x x (cid:19) , (18)whose iterates are equivalent to those of the nonlinear recurrence (15).12 emark 2. There is a more general notion of periodicity, due to Nakan-ishi [57], which extends Definition 2. This yields broad generalizations ofZamolodchikov’s Y-systems [66], a set of functional relations, arising fromthe thermodynamic Bethe ansatz for certain integrable quantum field the-ories, that were the prototype for the coefficient mutation (2) in a clusteralgebra. We shall introduce examples of generalized Y-systems in the sequel.Fordy and Marsh gave a complete classification of period 1 quivers. Theirresult can be paraphrased as follows.
Theorem 1.
Let ( a , . . . , a N − ) an ( N − -tuple of integers that is palin-dromic, i.e. a j = a N − j for all j ∈ [1 , N − . Then the skew-symmetricexchange matrix B = ( b ij ) with entries specified by b ,j +1 = a j and b i +1 ,j +1 = b ij + a i [ − a j ] + − a j [ − a i ] + , for all i, j ∈ [1 , N − , is cluster mutation-periodic with period 1, and everyperiod 1 skew-symmetric B arises in this way. The above result says that a period 1 skew-symmetric B matrix is com-pletely determined by the entries in its first row (or equivalently, its firstcolumn), and these form a palindrome after removing b . The entries a j in the palindrome are precisely the exponents that appear in the exchangerelation defining the cluster map ϕ , whose iterates are equivalent to thoseof the nonlinear recurrence relation x n x n + N = (cid:89) j : a j > x a j n + j + (cid:89) j : a j < x − a j n + j . (19)Thus (19) corresponds to a special sequence of mutations in a particularsubclass of cluster algebras. Such a nonlinear recurrence is an example of ageneralized T-system, in the terminology of [57].Next we would like to turn to the question of which recurrences of thisspecial type correspond to discrete integrable systems. We begin our ap-proach to this question in the next section, by considering the notion ofalgebraic entropy, which gives a measure of the growth of iterates in a dis-crete dynamical system defined by iteration of rational functions. There are various different ways of quantifying the growth, or complexity, ofa discrete dynamical system (see [1], for instance). In the context of discrete13ntegrability of birational maps, Bellon and Viallet introduced the conceptof algebraic entropy, and proposed that zero algebraic entropy should be acriterion for integrability [4]. For a birational map ϕ , one can calculate thedegree d n = deg ϕ n , given by the maximum of the degrees of the componentsof the map ϕ n , and then the algebraic entropy is defined to be E := lim n →∞ log d n n . Typically, the degree d n grows exponentially with n , so E >
0, but in rarecases there can be subexponential growth, leading to vanishing entropy. Inthe case of birational maps in two dimensions, the types of degree growthhave been fully classified [12], and there are only four possibilities: boundeddegrees, linear growth, quadratic growth, or exponential growth; the firstthree cases, with zero entropy, coincide with the existence of invariant folia-tions. Thus, at least for maps of the plane, the requirement of zero entropyidentifies symplectic maps that are integrable in the sense that they sat-isfy the conditions needed for a discrete analogue of the Liouville-Arnoldtheorem to hold [6, 53, 65].Measuring the degree growth and seeking maps with zero algebraic en-tropy is a useful tool for identifying discrete integrable systems. (For anotherapproach, based on the growth of heights in orbits defined over Q or a num-ber field, see [36].) Once such a map has been identified, it leaves open thequestion of Liouville integrability; this is discussed in the next section. Fornow, we concentrate on the case of maps arising from cluster algebras, andconsider algebraic entropy in that setting.The advantage of working with cluster maps is that, due to the Laurentproperty, it is sufficient to consider the growth of degrees of the denominatorsof the cluster variables in order to determine the algebraic entropy. Inparticular, in the period 1 case, by Proposition 1 every iterate of (19) canbe written in the form x n = P n ( x ) x d n , (20)where the polynomial P n is not divisible by any of the x j from the initialcluster, and the monomial x d n = (cid:81) Nj =1 x d ( j ) n j is specified by the integer vector d n = ( d (1) n , . . . , d ( N ) n ) T , known as a d-vector. From the fact that cluster variables are subtraction-free rational expressions in x (or, a fortiori, from the fact that these Laurentpolynomials are now known to have positive integer coefficients [35]), it14ollows that the d-vectors in a cluster algebra satisfy the max-plus tropicalanalogue of the exchange relations for the corresponding cluster variables[19, 24], where the latter is obtained from (4) by replacing each additionwith max, and each multiplication with addition. In the case of (19), thisimplies the following result. Proposition 2. If x n given by (20) satisfies (19), then the sequence ofvectors d n satisfies the tropical recurrence relation d n + d n + N = max (cid:88) j : a j > a j d n + j , − (cid:88) j : a j < a j d n + j . (21)Note that the equality in (21) holds componentwise. For a detailed proofof this result, see [26].In the period 1 situation, the problem of determining the evolution ofd-vectors can be simplified further, upon noting that the first component of d n has the initial values d (1)1 = − , d (1) j = 0 for 2 ≤ j ≤ N, (22)while each of the other components d ( k ) n for k ∈ [2 , N ] has the same set ofinitial values but shifted by k − d (2)1 = 0, d (2)2 = − d (2) j = 0 for 3 ≤ j ≤ N + 1, since the first division by the variable x appears in x N +2 , etc.). The total degree of the monomial x d n is thesum (cid:80) Nk =1 d ( k ) n of the components of the d-vector, and if the componentsare all non-negative then this coincides with the degree of the denominatorof the rational function (20). Unless there is periodicity of d-vectors, cor-responding to degrees remaining bounded (which can only happen in finitetype cases like Example 1), then all these components are positive for largeenough n . Moreover, it is not hard to see that the growth of the degree ofthe numerators P n appearing in the Laurent polynomials (20) is controlledby that of the denominators. Thus, to determine the growth of degrees ofLaurent polynomials generated by (19), it is sufficient to consider the solu-tion of the scalar version of (21), with initial data given by (22), and thegrowth of this determines the algebraic entropy. Example 9.
For the recurrence (7) in Example 2, the tropical equation fordetermining the degrees of d-vectors is given in scalar form by d n + d n +4 = max( d n +1 + d n +3 , . (23)15f we take initial values d = − d = d = d = 0, corresponding to (22),then by induction it follows that d n ≥ n ≥
2, so that the max onthe right-hand side of (23) can be replaced by its first entry, to yield thelinear recurrence d n + d n +4 = d n +1 + d n +3 for n ≥ . The characteristic polynomial of the latter factorizes as ( λ − ( λ + λ +1) =0, leading to the solution d n = n/ − √ ε n − ε − n ) , ε = ( − √ / . Thus we have a sequence that grows linearly with n , beginning with − , , , , , , , , , , , , , , , , . . . , where each positive integer appears three times in succession, which cor-responds to the degree of the denominator of x n in each of the variables x , x , x , x separately. Clearly the total degree of the denominator alsogrows linearly, and the algebraic entropy is lim n →∞ (log d n ) /n = 0 in thiscase. Example 10.
The exchange matrix (9) in Example 3 is period 2 rather thanperiod 1, but we can still calculate the growth of d-vectors in the recurrence(11) by taking its tropical version, namely d n + d n +3 = 2 max( d n +1 , d n +2 ) , and choosing the initial values d = − d = d = 0, which produces asequence beginning − , , , , , , , , , , , , . . . . By induction one can show that d n +2 ≥ d n +1 for n ≥
0, so in fact the linearrecurrence d n +3 + d n = 2 d n +2 holds for this sequence, with characteristic equation ( λ − λ − λ −
1) = 0,and it turns out that the differences F n = d n +3 − d n +2 > d n ∼ C (cid:32) √ (cid:33) n , and the algebraic entropy E = log((1 + √ / n →∞ (log log x n ) /n for the sequence (13) - see [39]. Example 11.
For the period 1 exchange matrix (14) in Example 4, weconsider the recurrence d n + d n +6 = max( d n +1 + d n +5 , d n +3 ) , (24)which is the max-plus analogue of (15), and take initial data d = − , d = d = · · · = d = 0 , (25)which generates a degree sequence beginning − , , , , , , , , , , , , , , , , , , , , , , , , , , , . . . . (26)In order to simplify the analysis of (24), we observe that the combination U n = d n +2 − d n +1 + d n (27)satisfies a recurrence of fourth order, namely U n +4 + 2 U n +3 + 3 U n +2 + 2 U n +1 + U n = max( U n +3 + 2 U n +2 + U n +1 , . (28)(The origin of the substitution (27) will be explained in the next section.)The values in (25) correspond to the initial conditions U = − , U = U = U = 0for (28), which generate a sequence ( U n ) beginning with − , , , , , − , , , − , , − , , , − , , , − , , − , , , − , , − , , , − , . . . , and further calculation with a computer shows that this sequence does notrepeat for the first 40 steps, but then U = − U = U = U = 0,so it is periodic with period 41. Thus in terms of the shift operator S , whichsends n → n + 1,( S − U n = ( S − d n +2 − d n +1 + d n ) = ( S − S − d n = 0 , λ = 1 as atriple root, and all other characteristic roots have modulus 1. Therefore, forsome constant C (cid:48) > d n ∼ C (cid:48) n as n → ∞ , which implies that (15) has algebraic entropy E = 0.The preceding examples indicate that we should regard (7) and (15)as being integrable in some sense, and (11) as non-integrable. Accordingto the relation (8), we know that (7) has at least one conserved quantity K ; and it turns out to have three independent conserved quantities [42].The recurrence (11) also has a conserved quantity, given by (12), but it ispossible to show that it can have no other algebraic conserved quantitites,independent of this one. In the next section we will derive two independentconserved quantities for (15), and we will discuss the interpretation of allthese examples from the viewpoint of Liouville integrability.In [26], a detailed analysis of the behaviour of the tropical recurrences(21) led to the conjecture that the algebraic entropy of (19) should be posi-tive if and only if the following condition holds:max N − (cid:88) j =1 [ a j ] + , − N − (cid:88) j =1 [ − a j ] + ≥ . (29)In other words, in order for the cluster map defined by (19) to have a zeroentropy, the degree of nonlinearity cannot be too large. The analysis ofalgebraic entropy for other types of cluster maps has been carried out morerecently using methods based on Newton polytopes [27], and using the samemethods it is also possible to prove the above conjecture . By enumeratingthe possible choices of exponents that lie below the bound (29), this leadsto a complete proof of a classification result for nonlinear recurrences of theform (19), as stated in [26]. Theorem 2.
A cluster map ϕ given by a recurrence (19) has algebraicentropy E = 0 if and only if it belongs to one of the following four families:(i) For even N = 2 m , recurrences of the form x n x n +2 m = x n + m + 1 . (30) P. Galashin, private communication, 2017 ii) For N ≥ and ≤ q ≤ (cid:98) N/ (cid:99) , recurrences of the form x n x n + N = x n + q x n + N − q + 1 . (31) (iii) For even N = 2 m and ≤ q ≤ m − , recurrences of the form x n x n +2 m = x n + q x n +2 m − q + x n + m . (32) (iv) For N ≥ and ≤ p < q ≤ (cid:98) N/ (cid:99) , recurrences of the form x n x n + N = x n + p x n + N − p + x n + q x n + N − q . (33)Case (i) is somewhat trivial: the recurrence (30) is equivalent to taking m copies of the Lyness 5-cycle x n x n +2 = x n +1 + 1 , for which every orbit has period 5, corresponding to the cluster algebra offinite type associated with the root system A ; so in this case the dynamicsis purely periodic and there is no degree growth. Both case (ii), which cor-responds to affine quivers of type ˜ A q,N − q , and case (iii) display linear degreegrowth, similar to Example 9. Case (iv) consists of Somos- N recurrences,which display quadratic degree growth [56], as in Example 11. Hence onlyzero, linear, quadratic or exponential growth is displayed by the cluster re-currences (19). Interestingly, these are the only types of growth found inthe other families of cluster maps considered in [27]. We do not know ifother types of growth are possible; are there cluster maps with cubic degreegrowth, for instance? So far we have alluded to the concept of integrability, but have skirtedaround the issue of giving a precise definition of what it means for a mapto be integrable. An expected feature of integrability is the ability to findexplicit solutions of the equations being considered; the recurrence (7) dis-plays this feature, because all of its iterates satisfy a linear recurrence ofthe form (8), which can be solved exactly. There are many other criteriathat can be imposed: existence of sufficiently many conserved quantities orsymmetries, or compatibility of an associated linear system (Lax pair), forinstance; and not all of these requirements may be appropriate in differentcircumstances. It is an unfortunate fact that the definition of an integrable19ystem varies depending on the context, i.e. whether it be autonomous ornon-autonomous ordinary differential equations, partial differential equa-tions, difference equations, maps or something else that is being considered.Thus we need to address this problem and clarify the context, in order tospecify what integrability means for maps associated with cluster algebras.There is a precise definition of Liouville integrability in the context offinite-dimensional Hamiltonian mechanics, on a real symplectic manifold M of dimension 2 m , with associated Poisson bracket { , } : given a particularfunction H , the Hamiltonian flow generated by H is completely integrable,in the sense of Liouville, if there exist m independent functions on M (in-cluding the Hamiltonian), say H = H , H , . . . , H m , which are in involutionwith respect to the Poisson bracket, i.e. { H j , H k } = 0 for all j, k . In the con-text of classical mechanics, this notion of integrability provides everythingone could hope for. To begin with, systems satisfying these requirementshave (at least) m independent conserved quantities: all of the first integrals H , . . . , H m are preserved by the time evolution, so each of the trajecto-ries lies on an m -dimensional intersection of level sets for these functions.Furthermore, Liouville proved that the solution of the equations of motionfor such systems can be reduced to a finite number of quadratures, so theyreally are “able to be integrated” as one would expect; and Arnold showedin addition that the flow reduces to quasiperiodic motion on compact m -dimensional level sets, which are diffeomorphic to tori T m [2], so nowadaysthe combined result is referred to as the Liouville-Arnold theorem. Anotherapproach to integrability is to require a sufficient number of symmetries,and this is a consequence of the Liouville definition: the Hamilton’s equa-tions arising from H have the maximum number of commuting symmetries,namely the flows generated by each of the first integrals H j .The notion of Liouville integrability can be extended to symplectic mapsin a natural way [6, 53, 65]. However, the requirement of working in evendimensions is too restrictive for our purposes, so instead of a symplectic formwe start with a (possibly degenerate) Poisson structure and consider Poissonmaps ϕ , defined in terms of the pullback of functions, given by ϕ ∗ F = F · ϕ . Definition 3.
Given a Poisson bracket { , } on a manifold M , a map ϕ : M → M is called a Poisson map if { ϕ ∗ F, ϕ ∗ G } = ϕ ∗ { F, G } holds for all functions F, G on M .(We are being deliberately vague about what sort of Poisson manifold( M, { , } ) is being considered, e.g. a real smooth manifold, or a complex20lgebraic variety, and what sort of functions, e.g. smooth/analytic/rational,because this may vary according to the context.)In order to have a suitable notion of integrability for cluster maps, wefirst require a compatible Poisson structure of some kind. In general, givena difference equation or map, there is no canonical way to find a compatiblePoisson bracket. Fortunately, it turns out that for cluster algebras there isoften a natural Poisson bracket, of log-canonical type, that is compatiblewith cluster mutations; and there is always a log-canonical presymplecticform [18, 29, 30, 47]. Example 12. Somos-5 Poisson bracket:
The skew-symmetric exchangematrix − − − − − −
11 0 − − − (34)is cluster mutation-periodic with period 1. Its associated cluster map is aSomos-5 recurrence, which belongs to family (iv) above, given by (33) with N = 5, p = 1, q = 2. The skew-symmetric matrix P = ( p ij ) given by P = − − − − − − − − − − defines a Poisson bracket, given in terms of the original cluster variables x = ( x , x , x , x , x ) by { x i , x j } = p ij x i x j . (35)This bracket is called log-canonical because it is just given by the constantmatrix P in terms of the logarithmic coordinates log x i . It is also compatiblewith the cluster algebra structure, in the sense that it remains log-canonicalunder the action of any mutation, i.e. writing x (cid:55)→ µ k ( x ) = x (cid:48) = ( x (cid:48) i ), inthe new cluster variables it takes the form { x (cid:48) i , x (cid:48) j } = p (cid:48) ij x (cid:48) i x (cid:48) j for some constant skew-symmetric matrix P (cid:48) = ( p (cid:48) ij ). Moreover, under thecluster map ϕ = ρ − · µ defined by ϕ : ( x , x , x , x , x ) (cid:55)→ (cid:18) x , x , x , x , x x + x x x (cid:19) , (36)21he bracket (35) is preserved, in the sense that for all i, j ∈ [1 ,
5] the pullbackof the coordinate functions by the map satisfies { ϕ ∗ x i , ϕ ∗ x j } = ϕ ∗ { x i , x j } . Hence ϕ is a Poisson map with respect to this bracket.Given a Poisson map, we can give a definition of discrete integrability,by adapting a definition from [64], that applies in the continuous case ofHamiltonian flows on Poisson manifolds. Definition 4.
Suppose that the Poisson tensor is of constant rank 2 m ona dense open subset of a Poisson manifold M of dimension N , and that thealgebra of Casimir functions is maximal, i.e. it contains N − m independentfunctions. A Poisson map ϕ : M → M is said to be completely integrableif it preserves N − m independent functions F , . . . , F N − m which are ininvolution, including the Casimirs. Example 13. Complete integrability of the ˜ A , cluster map: Setting P = B with the exchange matrix (6) in Example 2, the bracket { x i , x j } = b ij x i x j is compatible with the cluster algebra structure, and is preserved by thecluster map ϕ corresponding to (7). At points where all coordinates x j arenon-zero, the Poisson tensor has full rank 4, since B is invertible; so thereare no Casimirs. Note that B − = − B , so B is proportional to its owninverse, and the map ϕ is symplectic, i.e. ϕ ∗ ω = ω , where up to overallrescaling the symplectic form is ω = (cid:88) i The fact that cluster variables obtained from affine quivers sat-isfy linear relations with constant coefficients, such as (8), has been shownin various different ways: for type A in [25, 26], using Dodgson condensa-tion (equivalently, the Desnanot-Jacobi formula); for types A and D in thecontext of frieze relations [3]; and for all simply-laced types A , D , E in [49],using cluster categories (but see also [11, 61] for another family of quiversmade from products of finite and affine Dynkin types A ). The fact that23here are additional linear relations with periodic coefficients, like (38), wasshown for all ˜ A p,q quivers in [26], where it was also found that the quantities J i are coordinates in the dressing chain for Schr¨odinger operators, and thishas recently been extended to affine types D and E [60]. Example 14. Non-existence of a log-canonical bracket for ˜ A , : Forthe cluster algebra of type ˜ A , , defined by the skew-symmetric exchangematrix B = − − − it is easy to verify that there is no bracket of log-canonical form, like (35),that is compatible with cluster mutations. However, iterates of the clustermap, defined by the recurrence x n x n +3 = x n +1 x n +2 + 1 , satisfy the linear relation x n +4 − Kx n +2 + x n = 0, for a first integral K . Infact, setting u n = x n x n +1 yields a recurrence of second order, u n u n +2 = u n +1 ( u n +1 + 1) , (39)and, rewriting K in terms of u , u , this corresponds to a symplectic mapˆ ϕ in the ( u , u ) plane with symplectic form ˆ ω = d log u ∧ d log u and onefirst integral; so the map ˆ ϕ is completely integrable. Example 15. Casimirs for Somos-5: The Poisson tensor for the Somos-5 map (36), defined by (35), has rank 2 on C \ { x i = 0 } (away from thecoordinate hyperplanes). The kernel of the matrix P is spanned by thevectors˜ v = (1 , − , , , T , ˜ v = (0 , , − , , T , ˜ v = (0 , , , − , T , (40)which correspond to three independent Casimir functions F = x ˜ v = x x x , F = x ˜ v = x x x , F = x ˜ v = x x x , whose Poisson bracket with any other function G vanishes: { F j , G } = 0 for j = 1 , , 3. There are two independent first integrals H , H , i.e. functionsthat are preserved by the action of ϕ , so that ϕ ∗ H i = H i · ϕ = H i for i = 1 , F j [40]: H = F F F + 1 F + 1 F + 1 F + 1 F F F , (41) H = F F + F F + 1 F F + 1 F F + 1 F F F . (42)However, the full algebra of Casimirs is not preserved by the map ϕ , because F , F , F transform as ϕ ∗ F = F , ϕ ∗ F = F , ϕ ∗ F = F F + 1 F F F . Hence the Somos-5 map is not completely integrable with respect to thisbracket.The previous two examples show that if the exchange matrix B is de-generate, then the cluster coordinates may not be the correct ones to use,as either there is no invariant log-canonical bracket in these coordinates, asin the case of ˜ A , , or even if there is such a bracket, a full set of Casimirs isnot preserved by the cluster map. (A Poisson map sends Casimirs to otherCasimirs, but need not preserve each Casimir individually.) The way outof this quandary, which was already hinted at in Example 14, is to workon a reduced space where the map ϕ reduces to a symplectic map ˆ ϕ . Itturns out that there is a canonical way to do this, based on the presym-plectic form ω associated with the cluster algebra, which in general, for anyskew-symmetric exchange matrix B = ( b ij ), is given by the formula (37)above.In the case that B is nondegenerate (which is possible for even N only,as in Example 2), ω is a closed, nondegenerate 2-form, so the cluster mapis symplectic, but otherwise ω has a null distribution, generated by vectorfields of the form N (cid:88) j =1 w j x j ∂∂x j , for w = ( w j ) ∈ ker B. These vector fields all commute with other, and can be integrated to yielda commuting set of scaling symmetries: each w ∈ ker B generates a one-parameter scaling group x (cid:55)→ ˜ x = λ w · x , λ ∈ C ∗ , (43)25here the notation means that each component is scaled so that ˜ x j = λ w j x j .Regarding B as a linear transformation on Q N , skew-symmetry means thatthere is an orthogonal direct sum decomposition Q N = im B ⊕ ker B . If B has rank 2 m , then an integer basis v , v , . . . , v m for im B yields a completeset of rational functions invariant under the symmetries (43), given by themonomials u j = x v j , j = 1 , . . . , m. (44)In the case that B has period 1, it was shown in [26] that by choosing thebasis suitably, the rational map π : x (cid:55)→ u = ( u j ) reduces ϕ to a birationalsymplectic map ˆ ϕ in dimension 2 m , with symplectic form ˆ ω , in the sensethat ˆ ϕ · π = π · ϕ , and π ∗ ˆ ω = ω , whereˆ ω = (cid:88) i Given a cluster mutation-periodic skew-symmetric exchangematrix B with period 1, of rank 2 m , and the symplectic coordinates ( u j ) ∈ C m defined by (44) with a palindromic basis, the U-system is the recur-rence corresponding to the reduced cluster map ˆ ϕ , which, for some rationalfunction F , has the form u n u n +2 m = F ( u n +1 , . . . , u n +2 m − ) . (46)We have already seen an example of a U-system, namely the reduced re-currence (39) for ˜ A , . An integrable U-system corresponds to the canonicalversion of integrability for maps: the U-system is equivalent to a symplecticmap in dimension 2 m , so m independent first integrals in involution areneeded for complete integrability. 26 xample 16. Complete integrability of the Somos-5 U-system: With B given by (34), a palindromic basis for im B is written using (40) as v = ˜ v + ˜ v = (1 , − , − , , T , v = ˜ v + ˜ v = (0 , , − , − , T , so the reduced coordinates are u = x x x x = F F , u = x x x x = F F , and ω = (cid:80) i Setting β = 0 in (48) produces x n x n +6 = αx n +1 x n +5 + γx n +3 . (50)This differs from (15) and (18) by the inclusion of coefficients α, γ , whichcan be achieved by augmenting the cluster algebra with frozen variables thatappear in the exchange relations but do not themselves mutate (see [25] andreferences, for instance), and does not change other features such as Poissonbrackets or the (pre)symplectic forms. Upon applying the method in [45],we can obtain (48) as a plane wave reduction of (49), by setting T ( m , m , m ) = a m a m a m x n , n = m + 3 m + 2 m , with m arbitary, and taking α = a /a , γ = a /a . Under this reduc-tion, the linear system whose compatibility gives the discrete KP equationbecomes Y n ψ n +3 + αζψ n +2 = ξψ n ,ψ n +3 − X n ψ n +1 = ζψ n , (51)where ψ n is a wave function, ζ, ξ are spectral parameters, and X n = x n +2 x n +3 x n +4 x n +1 , Y n = x n +4 x n x n +3 x n +1 . The equation (50) is the compatibility condition for these two linear equa-tions for ψ n (to be precise, the parameter γ arises as an integration con-stant). This is more conveniently seen by writing the second linear equationin matrix form, with a vector Ψ n = ( ψ n , ψ n +1 , ψ n +2 ) T , asΨ n +1 = M n ψ n , M n = ζ X n , (52)28nd then using the second linear equation in (51) to reformulate the firstone as an eigenvalue problem with Ψ n as the eigenvector, that is L n Ψ n = ξ Ψ n , L n = ζY n u n ζαζ α ζ ( Y n +1 + αX n ) u n +1 ζu n +2 ζ α + u − n +1 ζ ( Y n +2 + αX n +1 ) . (53)In the above expression for the Lax matrix L n , we have introduced thequantities u n = X n Y n = x n x n +2 x n +1 , which for n = 1 , , , v = (1 , − , , , , T , v = (0 , , − , , , T , v =(0 , , , − , , T , v = (0 , , , , − , T for im B , with B as in (14), andsatisfy the U-system u n u n +4 = αu n +1 u n +2 u n +3 + γu n +1 u n +2 u n +3 (54)(which should be compared with the tropical formulae (27) and (28) above),correponding to the reduced cluster map ˆ ϕ . The symplectic form ˆ ω , suchthat ˆ ϕ ∗ ˆ ω = ˆ ω , isˆ ω = (cid:88) i Y-system is a set of difference equations arising as relations betweencoefficients appearing from a sequence of mutations in a cluster algebra withperiodicity. The original Y-systems were obtained by Zamolodchikov as aset of functional equations in certain quantum field theories associated withsimply-laced affine Lie algebras [66], yet they arise from cluster algebras offinite type obtained from the corresponding finite-dimensional root systems,and display purely periodic dynamics. Generalized Y-systems were definedby Nakanishi [57] starting from a general notion of periodicity in a clusteralgebra, and typically display complicated dynamical behaviour.Here we concentrate on the case of cluster mutation-periodic quivers withperiod 1, for which the Y-system can be written as a single scalar differenceequation, given by y n y n + N = (cid:81) N − j =1 (1 + y n + j ) [ a j ] + (cid:81) N − j =1 (1 + y − n + j ) [ − a j ] + , (56)where, as in Theorem 1, a j = b ,j +1 are the components of the palindromic( N − a j is positive; there is no loss of generality indoing so, due to the freedom to replace B → − B , but some signs are reversedcompared with [44] and [57].) In this context, the coefficient-free recurrence(19) that defines the cluster map is referred to as the T-system . It was firstobserved in [24] that there is a relation between the evolution of coefficients y under mutations (2) in a cluster algebra, and the evolution of clustervariables x due to the associated coefficient-free cluster mutations given by(4), which can be summarized by the slogan that “the T-system providesa solution of the Y-system.” In the case at hand, the precise statement isthat making the subsitution y n = N − (cid:89) j =1 x a j n + j (57)in (56) provides a solution of the Y-system whenever x n satisfies the coefficient-free T-system (19).Although the equations (19) and (56) are both of order N , there can bea discrepancy between the solutions of the T-system and the Y-system, inthe sense that the general solution of the former does not yield the generalsolution of the latter. This discrepancy is determined by the following result.31 roposition 3. Let x n satisfy the modified T-system x n x n + N = Z n N − (cid:89) j =1 x [ a j ] + n + j + N − (cid:89) j =1 x [ − a j ] + n + j . (58) Then the substitution (57) yields a solution of the Y-system (56) if and onlyif Z n satisfies the Z-system N − (cid:89) j =1 Z a j n + j = 1 . (59)Each iteration of the modified T-system (58) with non-autonomous coef-ficients evolving according to (59) preserves the presymplectic form given by(37) in terms of the entries of the exchange matrix B , and if B is degeneratewe can use a palindromic basis for im B to reduce this to a non-autonomousrecurrence in lower dimension that preserves the symplectic form (45). Definition 6. The pair of equations (58) and (59) is called the T z -system .The U z -system associated with (58) is given by (59) together with u n u n +2 m = Z n F ( u n +1 , . . . , u n +2 m − ) , where the rational function F is the same as in (46).We conclude this section with a couple of examples. Example 18. Somos-5 Y-system and q-Painlev´e II: The Y-systemassociated with the exchange matrix (34) is y n y n +5 = (1 + y n +1 )(1 + y n +4 )(1 + y − n +2 )(1 + y − n +3 ) , and (noting that on the right-hand side of the substitution (57) there is thefreedom to shift n → n + 1) the general solution of this can be written as y n = x n x n +3 x n +1 x n +2 , where x n satisfies the non-autonomous Somos-5 relation x n x n +5 = Z n ( x n +1 x n +4 + x n +2 x n +3 ) , with Z n Z n +3 Z n +1 Z n +2 = 1 . y n = u n and solve the third order Z-system for Z n to write the U z -system as a non-autonomous version of the QRT map(47), that is u n u n +2 = Z n (1 + u − n ) , with Z n = β n q n , β n +2 = β n . The latter is equivalent to a q-Painlev´e II equation identified in [50], havinga continuum limit to the Painlev´e II differential equation d udz = 2 u + zu + α. Example 19. A q-Somos-6 relation: The Y-system corresponding to(14) is y n y n +6 = (1 + y n +1 )(1 + y n +5 )(1 + y − n +3 ) . Its general solution can be written as y n = x n x n +4 y n +2 , where x n satisfies a q-Somos-6 relation given by x n x n +6 = Z n ( x n +1 x n +5 + x n +3 ) , with Z n = α ± q n ± , (60)with the solution of the fourth order Z-system Z n Z n +4 Z n +2 = 1being given in terms of quantities α ± and q ± that alternate with the parityof n . Alternatively, one can write y n = u n u n +1 u n +2 with u n satisfying a non-autonomous version of (54), that is u n u n +4 = Z n ( u n +1 u n +2 u n +3 + 1) u n +1 u n +2 u n +3 , with Z n as in (60). The latter should be regarded as a fourth order analogueof a discrete Painlev´e equation. 33 Conclusions We have just scratched the surface in this brief introduction to cluster al-gebras and discrete integrability. Among other important examples that wehave not described here, we would like to mention pentagram maps [31] andcluster integrable systems related to dimer models [14, 33]. 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