May 12, 2015 - The quiver which gives the discrete KdV equation is obtained ... I,II equations and their higher order analogues have been obtained in ...

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arXiv:1505.03067v1 [math-ph] 12 May 2015

Naoto Okubo Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan Abstract We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to qdiscrete Painlev´e equations. These cluster algebras are obtained from quivers with an infinite number of vertices or with the mutation-period property. We will also show that a suitable transformation of quivers corresponds to a reduction of the difference equation.

1

Introduction

In this article, we deal with cluster algebras, which were introduced by Fomin and Zelevinsky [1, 2]. A cluster algebra is a commutative ring described by cluster variables and coefficients. A generating set of the cluster algebra is defined by mutation, which is a transformation of a seed consisting of a set of cluster variables, coefficients, and a quiver. Cluster variables and coefficients obtained from a mutation of an initial seed satisfy some difference equations. It is known that cluster variables can satisfy the discrete KdV equation [4] and the Hirota-Miwa equation [5], when the initial seed includes suitable quivers with infinite vertices [6]. These quivers have the property that an infinite number of mutations gives a permutation of its vertices. This property is called ‘mutation-period’ and a quiver with this property is called a mutation-periodic quiver. Several results concerning mutation-periodic quivers have been reported in [3]. All the mutation-periodic quivers in which a permutation of its vertices is achieved by a single mutation have already obtained. The quiver which gives the discrete KdV equation is obtained from a transformation of the quiver of the Hirota-Miwa equation. This transformation corresponds to a reduction from the Hirota-Miwa equation to the discrete KdV equation. In this paper, we construct the cluster algebras whose variables and coefficients satisfy the discrete mKdV equation, the discrete Toda equation [7], and some q-discrete Painlev´e equations [8]. We introduce the quiver which generalizes the one that corresponds to the discrete KdV equation and the Hirota-Miwa equation. Quivers of q-Painlev´e I,II equations and their higher order analogues have been obtained in [6, 9]. We shall introduce the quivers for the q-Painlev´e III,VI equations, which are mutation-periodic and are obtained from transformations of quivers for the discrete KdV equation and the discrete mKdV equation.

2

Cluster algebras

In this section, we briefly explain the notion of cluster algebra which we use in the following sections. Let x = (x1 , x2 , . . . , xN ), y = (y1 , y2 , . . . , yN ) be N -tuple variables. Let Q be a quiver with N vertices. Consider the quiver whose vertices correspond to the cluster variables. We assume that the quiver does not have a loop (i −→ i) or a 2-cycle (i −→ j −→ i). Each xi is called a cluster variable and each yi is called a coefficient. The triple (Q, x, y) is called a seed. Let λi,j be the number of arrows from i to j of the quiver Q. We define λi,j for all 1 ≤ i, j ≤ N as λj,i = −λi,j . As a quiver Q does not have a loop, λi,i = 0. A mutation is a particular transformation of seeds. Definition 2.1 Let µk : (Q, x, y) 7−→ (Q′ , x′ , y ′ ) the quiver Q, defined as follows.

(k = 1, 2, . . . , N ) be the mutation at the vertex k of

• Q′ is a new quiver, obtained by three operations on the quiver Q. 1

1. For all (i, j) such that λi,k > 0, λk,j > 0, we add λi,k λk,j arrows from i to j. 2. If 2-cycles appear by the operation 1, we remove all of them. 3. We reverse the direction of all directed arrows which have edges at the vertex k. ′ • New coefficients y ′ = (y1′ , y2′ , . . . , yN ) are defined from Q and y as:

yk′ = yk−1 , yi′ = yi yk−1 + 1 yi′ yi′

−λk,i

λi,k

= yi (yk + 1) = yi

(λk,i > 0), (λk,i = 0).

• New cluster variables x′ = (x′1 , x′2 , . . . , x′N ) are defined from Q, y and x as: Y Y 1 λ λ xj k,j + yk x′k = xj j,k , (yk + 1)xk λk,j >0

x′i = xi

(2.1)

(λi,k > 0),

λj,k >0

(2.2)

(i 6= k).

A mutation µk denotes the mutation at k or xk . For any seed (Q, x, y), it holds that µ2k (Q, x, y) = (Q, x, y). For any seed (Q, x, y) and (i, j) such that λi,j = 0, it holds that µi µj (Q, x, y) = µj µi (Q, x, y). Definition 2.2 Let us fix a seed (Q, x, y). This seed is called an initial seed. Let A(Q, x, y) be a cluster algebra (with coefficients) defined as A(Q, x, y) = Z(y)[x|x ∈ X] ⊂ Q(y)(x),

(2.3)

where X is the set of all the cluster variables obtained from iterative mutations to the initial seed. Now we define the coefficient-free cluster algebra as the pair of a quiver and cluster variables (Q, x), which is also called a seed, and by using its mutation. Definition 2.3 Let µk : (Q, x) 7−→ (Q′ , x′ )

(k = 1, 2, . . . , N ) be the mutation defined as follows.

• The definition of a new quiver Q′ is the same as that in Definition 2.1. • Let x′ = (x′1 , x′2 , . . . , x′N ) be the new cluster variables defined by Q and x as: Y Y 1 λ λ xj k,j + xj j,k , x′k = xk λk,j >0

x′i = xi

λj,k >0

(2.4)

(i 6= k).

Definition 2.4 Let us fix an initial seed (Q, x). We define a coefficient-free cluster algebra A(Q, x) as A(Q, x) = Z[x|x ∈ X] ⊂ Q(x),

(2.5)

where X is the set of cluster variables obtained from a iteration of all mutations to the initial seed.

3

Bilinear equations satisfied by cluster variables and their reductions

In this section, we show that cluster variables satisfy bilinear equations related to discrete integrable systems, if the initial seed includes suitable quivers with infinite vertices. These quivers are obtained from a transformation of quivers which corresponds to a reduction of a certain difference equations. 2

Figure 1: The dKdV quiver (The numbers on vertices correspond to cluster variables.)

3.1

The discrete KdV equation and the Hirota-Miwa equation

We construct cluster algebras whose cluster variables satisfy the discrete KdV equation and the HirotaMiwa equation. The quivers of the initial seeds were obtained in [6]. We consider quivers of general form of the discrete KdV equation and the Hirota-Miwa equation. First we consider coefficient-free cluster algebras. For any N, M ≥ 1, let QN,M KdV be a quiver as shown in Figure 1 (the dKdV quiver), where the numbers attached to the arrows of the quivers denote the numbers of arrows pointing in the same direction. Note that each vertex (n, m) corresponds to a cluster variable xm n . Let x be the set ′ of these cluster variables. We take (QN,M , x) as an initial seed. We define µ as the iteration of all i KdV mutations at the vertices i in Figure 2. Figure 2 shows the case where N ≥ M . Note that mutations at vertices with the same numbers are commutative. We apply the mutation to the initial seed in the order µ′1 , µ′2 , . . . , µ′max[N,M] , µ′1 , µ′2 , . . . . The new cluster variable obtained by mutation at xm n is denoted m+1 by xn+2 . We then obtain the following proposition from the definition of mutation (2.4). Proposition 3.1 Consider the coefficient-free cluster algebra A(QN,M KdV , x). For any n, m ∈ Z, the cluster variables xm n satisfy the bilinear equation m+1 m m m+1 m xn . xm+1 n+1 xn−1 = xn−1 xn+1 + xn

(3.1)

This equation is nothing but the bilinear form of the discrete KdV equation [4]. For any N ≥ 1, let QN HM be the quiver as shown in Figure 3 (the HM quiver). Vertices 1 and N contained in a circle correspond to the cluster variables x00,0 and x00,−1 respectively, and vertices N + 1 and 2 contained in a square correspond to the cluster variables x0N,0 and x0N,1 respectively. For the other m+1 m m the one above vertices adjacent to xm n,l , we denote by xn+1,l the one on the right of xn,l , and by xn,l m N xn,l , and so on. Let x be these cluster variables. We take (QHM , x) as an initial seed and mutate it in the order µ′1 , µ′2 , . . . , µ′N +1 , µ′1 , µ′2 , . . . . We denote by xm+1 n+1,l−1 the new cluster variable obtained by mutation at xm n,l . Then we obtain the following proposition from the definition of mutation (2.4). Proposition 3.2 Consider the coefficient-free cluster algebra A(QN HM , x). For any n, m, l ∈ Z, the cluster variables xm satisfy the bilinear equation n,l m+1 m m+1 m m xm+1 n+1,l xn,l+1 = xn,l+1 xn+1,l + xn,l xn+1,l+1 .

3

(3.2)

Figure 2: The dKdV quiver (The numbers at the vertices denote the order of mutations.)

Figure 3: The HM quiver (The numbers at the vertices denote the order of mutations. Vertices 1 and N contained in a circle correspond to the cluster variables x00,0 and x00,−1 respectively, and vertices N + 1 and 2 contained in a square correspond to the cluster variables x0N,0 and x0N,1 respectively.)

4

Figure 4: Reduction from the HM quiver to the dKdV quiver (Superposition of vertices with same numbers.) This equation is the Hirota-Miwa equation [5]. The quiver of the discrete KdV equation can be obtained from a certain transformation of the quiver 1,1 of the Hirota-Miwa equation. In particular, we show that the dKdV quiver QKdV is obtained from the HM quiver Q2HM by applying the following two operations successively on the quiver Q2HM . 1. Among the arrows from (n, m, l) + k(1, 0, 1) to (n′ , m′ , l′ ) + k(1, 0, 1) (k ∈ Z), we remove all the arrows with k 6= 0 in Q2HM (cf. Figure 4). 2. We superimpose the vertices (n, m, l) + k(1, 0, 1) (k ∈ Z) on the vertex (n, m, l). (In Figure 4, we superimpose the vertices with the same number.) The dKdV quiver Q1,1 KdV is obtained from the above operation, which will be called a (1, 0, 1)-reduction of a quiver. In a similar way the (a, b, c)-reduction of a quiver is defined. In fact, reduction of a quiver corresponds to reduction of a difference equation. In this case, the discrete KdV equation (3.1) is obtained m m m from the Hirota-Miwa equation (3.2) by imposing the reduction condition xm n+1,l+1 = xn,l and xn := xn,0 .

3.2

The discrete mKdV equation and the discrete Toda equation

We construct cluster algebras whose cluster variables satisfy the discrete mKdV equation and the discrete Toda equation by the reduction of the quiver of the Hirota-Miwa equation. Let QmKdV (the dmKdV quiver) be the quiver obtained from the (0, 0, 2)-reduction of the HM quiver Q1HM (cf. Figures 5, 6). In Figure 6, the numbers at vertices denote the order of mutations. Vertices 1 in a circle and a square correspond to the cluster variables w00 and x00 respectively. For the other vertices adjacent to wnm and m m m m m+1 xm , and n , we denote by wn+1 and xn+1 those on the right of wn , and xn respectively, and by wn m+1 m m xn those above wn and xn , and so on. Let x be these cluster variables. We take (QmKdV , x) as an m+1 initial seed and mutate it in the order µ′1 , µ′2 , µ′1 , µ′2 , . . . . We denote by xm+1 n+1 and wn+1 the new cluster m m variable obtained by mutation at wn and xn respectively. Then we obtain the following proposition by the definition of mutation (2.4). Proposition 3.3 Consider the coefficient-free cluster algebra A(QmKdV , x). For any n, m ∈ Z, the cluster variables wnm , xm n satisfy the bilinear equations: m+1 m m wn+1 xn = xm+1 wn+1 + wnm+1 xm n n+1 , m m+1 m m xm+1 xn+1 + xm+1 wn+1 . n n+1 wn = wn

(3.3)

Note that nonautonomous bilinear equations can be obtained from cluster algebras with coefficients. Consider the cluster algebra with coefficients A(QmKdV , x, y). Let the cluster variables wnm , xm n be defined as in the coefficient-free case. By the definition of a mutation (2.2), the cluster variables wnm , xm n (n, m ∈ Z)

5

Figure 5: Reduction from the HM quiver to the dmKdV quiver

Figure 6: The dmKdV quiver

satisfy the following bilinear equations: m+1 m m+1 m m+1 m wn+1 xn = am wn+1 + bm xn+1 , n xn n wn m m+1 m m+1 m m xm+1 wn+1 , xn+1 + dm n xn n+1 wn = cn wn

(3.4)

m m m m m where am n , bn , cn , dn are rational functions of the coefficients of the initial seed for which an + bn = m m cn + dn = 1 holds. These equations are the bilinear form of the discrete mKdV equation. The discrete mKdV equation (3.3) is obtained from the Hirota-Miwa equation (3.2) by imposing the m m m m m reduction condition xm n,l+2 = xn,l and wn := xn,0 , xn := xn,1 . This reduction of the difference equation corresponds to the (0, 0, 2)-reduction of the HM quiver. Let QN,M mKdV be the quiver as shown in Figure 7, which shows the case of N ≥ M . This quiver is a generalization of the dmKdV quiver. In fact, QmKdV = Q1,1 mKdV . In Figure 7, numbers at vertices denote the order of mutations. The discrete mKdV equations (3.3) and (3.4) are obtained from the generalized dmKdV quiver in the same way. Let QT (the dToda quiver) be the quiver obtained from the (1, −1, 1)-reduction of the HM quiver Q1HM (cf. Figures 8, 9). In Figure 9, the numbers at vertices denote the order of mutations. The vertex 1 contained in a circle corresponds to the cluster variable x00 . For the other vertices adjacent to xm n , we m m+1 denote by xm that above xm n+1 the one on the right of xn , and by xn n , and so on. Let x be these cluster variables. We take (QT , x) as an initial seed and mutate it in the order µ′1 , µ′2 , µ′1 , µ′2 , . . . . We denote by m xm n+2 the new cluster variable obtained by mutation at xn . Then we obtain the following proposition by the definition of mutation (2.4).

Proposition 3.4 Consider the coefficient-free cluster algebra A(QT , x). For any n, m ∈ Z, the cluster variables xm n satisfy the bilinear equation m+1 m−1 m m 2 xm n+1 xn−1 = xn−1 xn+1 + (xn ) .

(3.5)

This equation is the bilinear form of the discrete Toda equation. The discrete Toda equation (3.5) is m obtained from the Hirota-Miwa equation (3.2) by imposing the reduction condition xm−1 n+1,l+1 = xn,l and m m xn := xn,0 . This reduction of the difference equation corresponds to the (1, −1, 1)-reduction of the HM quiver.

4

q-discrete Painlev´ e equations satisfied by coefficients

In this section, we show that coefficients in cluster algebras can satisfy q-discrete Painlev´e equations, if the initial seed includes suitable quivers with the mutation-period property. Quivers for the q-Painlev´e 6

Figure 7: Generalized dmKdV quiver

Figure 8: Reduction from the HM quiver to the dToda quiver

7

Figure 9: The dToda quiver

Figure 10: Reduction from the dKdV quiver to the q-PI quiver

Figure 11: The q-PI quiver

I,II equations have been obtained in [6, 9]. In this paper, we introduce the quivers for the q-Painlev´e III,VI equations in a similar way. We shall show that these quivers are obtained as reductions of the dKdV quiver and the dmKdV quiver.

4.1

Mutation-periodic quivers

The quivers of q-discrete Painlev´e equations have the property that mutations of their quivers are equal to permutation of the vertices. This is the so-called ‘mutation-period’ property, which we define as follows. Let Q be a quiver. For i = (i1 , i2 , . . . , ih ) (ij ∈ {1, 2, . . . , N }), we define an iteration of mutations µi as µi (Q) = µih µih−1 · · · µi1 (Q), where h is the number of applications of the mutation. For a permutation ν ∈ SN , let ν(Q) be the quiver in which we substituted vertices i for ν(i) in Q. Definition 4.1 i is a ν-period of Q if µi (Q) = ν(Q) holds. Q is said to be a mutation-periodic quiver if i and ν ∈ SN , as defined above, exist. In the case of h = 1 (i = (i1 )), all mutation-periodic quivers have already been obtained [3]. Quivers of q-discrete Painlev´e equations are mutation-periodic quivers. These quivers arise from a reduction of the dKdV quiver and the dmKdV quiver. We consider both cluster algebras with coefficients and coefficient-free cluster algebras.

4.2

The q-Painlev´ e I equation

We construct the quiver of the q-Painlev´e I equation by a reduction of the dKdV quiver. Let QPI (the q-PI quiver) be the quiver obtained from the (2, −1)-reduction of the dKdV quiver Q2,1 KdV (cf. Figures 10, 11). Let ν ∈ S4 be ν : (1, 2, 3, 4) 7→ (2, 3, 4, 1). i = (1) is a ν-period of QPI . Note that each vertex i corresponds to a cluster variable xi and a coefficient yi,1 . We take (QPI , x, y) as an initial seed, where x = (x1 , x2 , x3 , x4 ), y = (y1,1 , y2,1 , y3,1 , y4,1 ) and we mutate the initial seed in the order µi = µ1 , µν(i) = µ2 , µν 2 (i) = µ3 , . . . . The new cluster variables and the new coefficients are denoted as xn → xn+4 , yn,m → yn,m+1 . We put yn := ym,n (n ≡ m (mod 4)) and obtain the following seeds: µ4

· · · ←→ µ1

←→

(QPI ; x1 , x2 , x3 , x4 ; y1 , y2,1 , y3,1 , y4,1 ) (ν(QPI ); x5 , x2 , x3 , x4 ; y1,2 , y2 , y3,2 , y4,2 )

µ2

←→ (ν 2 (QPI ); x5 , x6 , x3 , x4 ; y1,3 , y2,3 , y3 , y4,3 ) µ3

(4.1)

3

←→ (ν (QPI ); x5 , x6 , x7 , x4 ; y1,4 , y2,4 , y3,4 , y4 ) µ4

←→

(QPI ; x5 , x6 , x7 , x8 ; y5 , y2,5 , y3,5 , y4,5 )

µ1

←→ · · · .

The following proposition is obtained from the definition of mutation (2.4). Proposition 4.2 Consider the coefficient-free cluster algebra A(QPI , x). For any n ∈ Z, the cluster variables xn satisfy the bilinear equation xn+4 xn = x2n+2 + xn+3 xn+1 .

(4.2)

The bilinear equation (4.2) can be obtained from the discrete KdV equation (3.1) by imposing the m 0 reduction condition xm−1 n+2 = xn and xn := xn . This reduction of the difference equation corresponds 8

to the (2, −1)-reduction of the dKdV quiver. It turns out that the corresponding coefficients satisfy the q-Painlev´e I equation. Theorem 4.3 Consider the cluster algebra with coefficient A(QPI , x, y). For any n ∈ Z, the coefficients yn satisfy the equation yn + 1 , (4.3) yn+1 yn−1 = c2 cn1 yn2 where c1 , c2 are the conserved quantities −1 yn+3 yn+1 +1 , c1 = −1 yn yn+2 +1

c2 =

2 yn+2 yn+1 yn −(n+1) c yn+1 + 1 1

(4.4)

and do not depend on n. This equation (4.3) is the q-Painlev´e I equation [10]. The q-Painlev´e I equation (4.3) is obtain from the bilinear equation (4.2) by the following transformation of variables: yn =

xn+2 xn . x2n+1

(4.5)

Proof By the definition of mutation (2.1), the coefficients yn,m satisfy yn = yn,n−1 (yn−1 + 1), −2 −1 yn,n−1 = yn,n−2 yn−2 +1 , yn,n−2 = yn,n−3 (yn−3 + 1),

(4.6)

−1 yn,n−3 = yn−4 ,

where we consider the index n of coefficients yn,m as n ∈ Z/4Z. We then obtain an equation only for yn : yn+4 =

(yn+3 + 1)(yn+1 + 1) . 2 −1 yn+2 + 1 yn

(4.7)

We put −1 +1 yn+3 yn+1 , un := −1 +1 yn yn+2

vn :=

2 yn+2 yn+1 yn , yn+1 + 1

(4.8)

and find un+1 = un from (4.7). Hence we obtain the conserved quantity un = c1 . Similarly, we obtain −(n+1) vn+1 = c1 vn from un = c1 and we obtain vn = c2 cn+1 , where c2 = vn c1 is also a conserved quantity. 1 n+1 We obtain the q-Painlev´e I equation (4.3) from vn = c2 c1 .

4.3

The q-Painlev´ e II equation

We construct the quiver of the q-Painlev´e II equation by a reduction of dKdV quiver. Let QPII (the 3,1 q-PII quiver) be the quiver obtained from the (3, −1)-reduction of the dKdV quiver QKdV (cf. Figures 12, 13). Let ν ∈ S5 be ν : (1, 2, 3, 4, 5) 7→ (2, 3, 4, 5, 1). i = (1) is a ν-period of QPII . Note that each vertex i corresponds to a cluster variable xi and a coefficient yi,1 . We take (QPII , x, y) as an initial seed, where x = (x1 , x2 , x3 , x4 , x5 ), y = (y1,1 , y2,1 , y3,1 , y4,1 , y5,1 ) and mutate the initial seed in the order µi , µν(i) , µν 2 (i) , . . . . The new cluster variables and the new coefficients are denoted as xn → xn+5 , yn,m → yn,m+1 . We put yn := ym,n (n ≡ m (mod 5)) and obtain the following seeds: µ5

· · · ←→ µ1

←→ µ2

←→

(QPII ; x1 , x2 , x3 , x4 , x5 ; y1 , y2,1 , y3,1 , y4,1 , y5,1 ) (4.9)

(ν(QPII ); x6 , x2 , x3 , x4 , x5 ; y1,2 , y2 , y3,2 , y4,2 , y5,2 ) µ3

2

(ν (QPII ); x6 , x7 , x3 , x4 , x5 ; y1,3 , y2,3 , y3 , y4,3 , y5,3 ) ←→ · · · .

We obtain the following proposition from the definition of mutation (2.4). 9

Figure 12: Reduction from the dKdV quiver to the q-PII quiver

Figure 13: The q-PII quiver

Proposition 4.4 Consider the coefficient-free cluster algebra A(QPII , x). For any n ∈ Z, the cluster variables xn satisfy the bilinear equation xn+5 xn = xn+3 xn+2 + xn+4 xn+1 .

(4.10)

The bilinear equation (4.10) is obtained from the discrete KdV equation (3.1) by imposing the reduction m 0 condition xm−1 n+3 = xn and xn := xn . This reduction of the difference equation corresponds to the (3, −1)-reduction of the dKdV quiver. Its coefficients satisfy the q-Painlev´e II equation. Theorem 4.5 Consider the cluster algebra with coefficient A(QPII , x, y). For any n ∈ Z, the coefficients yn satisfy the equation (−1)n n yn + 1 c1 yn+1 yn−1 = c2 c3 , (4.11) yn where c1 , c2 , c3 are the conserved quantities −1 2 2 +1 yn+4 yn+1 y2n+3 y2n+2 y2n+1 y2n −(4n+3) 2 c1 , , c22 = c1 = −1 (y + 1)(y yn yn+3 + 1 2n+2 2n+1 + 1) (4.12) y2n+3 (y2n+1 + 1) −1 2 c c3 = y2n (y2n+2 + 1) 1 and do not depend on n. We put fn := y2n , gn := y2n+1 and obtain gn + 1 , gn fn + 1 = c2 c3 c2n 1 fn

2n+1 fn+1 fn = c2 c−1 3 c1

gn gn−1

(4.13)

from (4.11). This equation is the q-Painlev´e II equation [11]. The q-Painlev´e II equation (4.11) is obtained from the bilinear equation (4.10) by the following transformation of variables: yn =

xn+3 xn . xn+2 xn+1

(4.14)

Proof By the definition of mutation (2.1), the coefficients yn,m satisfy yn = yn,n−1 (yn−1 + 1), −1 −1 yn,n−1 = yn,n−2 yn−2 +1 , −1 −1 yn,n−2 = yn,n−3 yn−3 +1 , yn,n−3 = yn,n−4 (yn−4 + 1),

(4.15)

−1 yn,n−4 = yn−5 ,

where we think of the index n of the coefficients yn,m as n ∈ Z/5Z. We obtain an equation only for yn : yn+5 =

(yn+4 + 1)(yn+1 + 1) −1 . −1 yn+3 + 1 yn+2 + 1 yn 10

(4.16)

Figure 14: Reduction from the dmKdV quiver to the q-PIII quiver

Figure 15: The q-PIII quiver

We put −1 yn+4 yn+1 +1 , un := −1 yn yn+3 +1

vn :=

yn+2 yn , −1 yn+1 +1

(4.17)

and find un+1 = un from (4.16). Hence, we obtain the conserved quantity un = c21 . Similarly, we obtain (−1)n+1 n+1 c1 ;

vn+2 = c21 vn from un = c21 , and vn = c2 c3

−(4n+3)

c22 = v2n+1 v2n c1

−1 −1 and c23 = v2n+1 v2n c1 are (−1)n+1 n+1 c1 .

conserved quantities. We then obtain the q-Painlev´e II equation (4.11) from vn = c2 c3

4.4

The q-Painlev´ e III equation

We construct the quiver of the q-Painlev´e III equation by a reduction of the dmKdV quiver. Let QPIII (the q-PIII quiver) be the quiver obtained from the (2, −1)-reduction of the dmKdV quiver Q2,1 mKdV (cf. Figures 14, 15). Let ν ∈ S6 be ν : (1, 2, 3, 4, 5, 6) 7→ (3, 4, 5, 6, 2, 1). i = (1, 2) is a ν-period of QPIII . Note that vertices (1, 2, 3, 4, 5, 6) correspond to cluster variables x = (w1 , x1 , w2 , x2 , w3 , x3 ) and coefficients y = (y1,1 , z1,1 , y2,1 , z2,1 , y3,1 , z3,1 ). We take (QPIII , x, y) as an initial seed and mutate it in the order µi = µ2 µ1 , µν(i) = µ4 µ3 , µν 2 (i) = µ6 µ5 , . . . . The new cluster variables are denoted as wn → xn+3 , xn → wn+3 and the new coefficients are denoted as yn,m → zn,m+1 , zn,m → yn,m+1 (if the coefficient corresponds to the vertex to which the mutation is applied) or yn,m → yn,m+1 , zn,m → zn,m+1 (otherwise). We put yn := ym,n , zn := zm,n (n ≡ m (mod 3)) and obtain the following seeds: µ5 µ6

· · · ←→ µ2 µ1

←→

(QPIII ; w1 , x1 , w2 , x2 , w3 , x3 ; y1 , z1 , y2,1 , z2,1 , y3,1 , z3,1 ) (ν(QPIII ); x4 , w4 , w2 , x2 , w3 , x3 ; z1,2 , y1,2 , y2 , z2 , y3,2 , z3,2 )

µ4 µ3

←→ (ν 2 (QPIII ); x4 , w4 , x5 , w5 , w3 , x3 ; z1,3 , y1,3 , z2,3 , y2,3 , y3 , z3 ) µ6 µ5

←→ µ1 µ2

←→

(4.18)

(QPIII ; x4 , w4 , x5 , w5 , x6 , w6 ; z4 , y4 , z2,4 , y2,4 , z3,4 , y3,4 ) µ3 µ4

(ν(QIII ); w7 , x7 , x5 , w5 , x6 , w6 ; y1,5 , z1,5 , z5 , y5 , z3,5 , y3,5 ) ←→ · · · .

We obtain the following proposition from the definition of mutation (2.4). Proposition 4.6 Consider the coefficient-free cluster algebra A(QPIII , x). For any n ∈ Z, the cluster variables wn , xn satisfy the bilinear equations: wn+3 xn = xn+2 wn+1 + wn+2 xn+1 , xn+3 wn = wn+2 xn+1 + xn+2 wn+1 .

(4.19)

The bilinear equations (4.19) are obtained from the discrete mKdV equation (3.3) by imposing the m−1 m 0 0 reduction condition wn+2 = wnm , xm−1 n+2 = xn and wn := wn , xn := xn . This reduction of the difference equation corresponds to the (2, −1)-reduction of the dmKdV quiver. In this case, the coefficients satisfy the q-Painlev´e III equation.

11

Theorem 4.7 Consider the cluster algebra with coefficient A(QPIII , x, y). For any n ∈ Z, the coefficients yn , zn satisfy the equations: yn+1 yn−1 = c2 c23 c2n 1

yn + 1 , (−1)n n c1 y n y n + c3 c4

2 2n zn+1 zn−1 = c−1 2 c3 c1

zn + 1

(−1)n n c1

z n z n + c3 c4

(4.20)

,

where c1 , c2 , c3 , c4 are conserved quantities c21 c23

yn+2 zn+2 = , yn zn =

2 −1 +1 yn+2 yn zn+1 = 2 , −1 +1 zn+2 zn yn+1 y2n z2n c24 = c1 y2n+1 z2n+1 c22

−(4n+1) y2n+1 z2n+1 y2n z2n c1 ,

(4.21)

and do not depend on n. If we put fn := y2n , gn := y2n+1 , we obtain gn + 1 2n+1 , gn gn + c3 c−1 4 c1 fn + 1 = c2 c23 c4n 1 fn (fn + c3 c4 c2n 1 )

fn+1 fn = c2 c23 c4n+2 1 gn gn−1

(4.22)

from (4.20). These equations are the q-Painlev´e III equation [12]. Proof By the definition of mutation (2.1), the coefficients yn,m , zn,m satisfy −1 −1 yn = yn,n−1 (yn−1 + 1) zn−1 +1 , −1 −1 yn,n−1 = yn,n−2 yn−2 +1 (zn−2 + 1), −1 yn,n−2 = zn−3 ,

−1 −1 zn = zn,n−1 (zn−1 + 1) yn−1 +1 , −1 −1 zn,n−1 = zn,n−2 zn−2 + 1 (yn−2 + 1),

(4.23)

−1 zn,n−2 = yn−3 ,

where we consider the index n of the coefficients yn,m , zn,m as n ∈ Z/3Z. The equations, only for yn , zn , are (yn+2 + 1)(zn+1 + 1) −1 , yn+3 = −1 zn+2 + 1 yn+1 + 1 zn (4.24) (zn+2 + 1)(yn+1 + 1) −1 . zn+3 = −1 yn+2 + 1 zn+1 + 1 yn If we put −1 yn+2 zn+1 +1 , un := −1 zn yn+1 +1

−1 zn+2 yn+1 +1 , vn := −1 yn zn+1 +1

tn := yn zn ,

(4.25)

we find un+1 = un , vn+1 = vn from (4.24), and obtain conserved quantities un = c1 c2 , vn = c1 c−1 2 . We (−1)n n −(4n+1) also obtain tn+2 = c21 tn from un vn = c21 , and tn = c3 c4 c1 . c23 = t2n+1 t2n c1 and c24 = t−1 2n+1 t2n c1 (−1)n n are conserved quantities. We obtain the q-Painlev´e III equation (4.20) from un = c1 c2 and tn = c3 c4 c1 by the elimination of zn . The equation for zn is the same as that for yn .

12

Figure 16: Reduction from the dmKdV quiver to the q-PVI quiver

4.5

Figure 17: The q-PVI quiver

The q-Painlev´ e VI equation

We now construct the quiver of the q-Painlev´e VI equation by a reduction of the dmKdV quiver. Let QPVI (the q-PVI quiver) be the quiver obtained from the (2, −2)-reduction of the dmKdV quiver Q1,1 mKdV (cf. Figures 16, 17). Let ν ∈ S8 be ν : (1, 2, 3, 4, 5, 6, 7, 8) 7→ (5, 6, 7, 8, 4, 3, 2, 1). i = (1, 2, 3, 4) is a ν-period of QPVI . Note that vertices (1, 2, . . . , 8) correspond to cluster variables x = (w1 , x1 , W1 , X1 , w2 , x2 , W2 , X2 ) and coefficients y = (y1,1 , z1,1 , Y1,1 , Z1,1 , y2,1 , z2,1 , Y2,1 , Z2,1 ). We take (QPVI , x, y) as an initial seed and mutate it in the order µi , µν(i) , µν 2 (i) , . . . . The new cluster variables are denoted as wn → Xn+2 , xn → Wn+2 , Wn → xn+2 , Zn → wn+2 and the new coefficients are denoted as yn,m → Zn,m+1 , zn,m → Yn,m+1 , Yn,m → zn,m+1 , Zn,m → yn,m+1 (in case the coefficient corresponds to the vertex to which the mutation is applied) or yn,m → yn,m+1 , zn,m → zn,m+1 , Yn,m → Yn,m+1 , Zn,m → Zn,m+1 (otherwise). We put yn := ym,n , zn := zm,n , Yn := Ym,n , Zn := Zm,n (n ≡ m (mod 2)) and obtain the following seeds: ···

µ5 µ6 µ7 µ8

←→

(QVI ; w1 , x1 , W1 , X1 , w2 , x2 , W2 , X2 ; y1 , z1 , Y1 , Z1 , y2,1 , z2,1 , Y2,1 , Z2,1 )

µ4 µ3 µ2 µ1

←→

(ν(QVI ); X3 , W3 , x3 , w3 , w2 , x2 , W2 , X2 ; Z1,2 , Y1,2 , z1,2 , y1,2 , y2 , z2 , Y2 , Z2 )

µ8 µ7 µ6 µ5

←→

(4.26)

(QVI ; X3 , W3 , x3 , w3 , X4 , W4 , x4 , w4 ; Z3 , Y3 , z3 , y3 , Z2,3 , Y2,3 , z2,3 , y2,3 )

µ1 µ2 µ3 µ4

←→

(ν(QVI ); w5 , x5 , W5 , X5 , X4 , W4 , x4 , w4 ; y1,4 , z1,4 , Y1,4 , Z1,4 , Z4 , Y4 , z4 , y4 )

µ5 µ6 µ7 µ8

←→

··· .

We obtain the following proposition from the definition of mutation (2.4). Proposition 4.8 Consider the coefficient-free cluster algebra A(QPVI , x). For any n ∈ Z, the cluster variables wn , xn , Wn , Xn satisfy the bilinear equations: wn+2 Xn = xn+1 Wn+1 + wn+1 Xn+1 , xn+2 Wn = wn+1 Xn+1 + xn+1 Wn+1 , Wn+2 xn = Xn+1 wn+1 + Wn+1 xn+1 ,

(4.27)

Xn+2 wn = Wn+1 xn+1 + Xn+1 wn+1 . The bilinear equations (4.27) are obtained from the discrete mKdV equation (3.3) by imposing the m−2 m 0 0 1 1 reduction condition wn+2 = wnm , xm−2 n+2 = xn and wn := wn , xn := xn , Wn := wn−1 , Xn := xn−1 . This reduction of the difference equation corresponds to the (2, −2)-reduction of the dmKdV quiver. Moreover, the coefficients satisfy the q-Painlev´e VI equation.

13

Theorem 4.9 Consider the cluster algebra with coefficient A(QPVI , x, y). For any n ∈ Z, the coefficients yn , zn , Yn , Zn satisfy the equations: (−1)n+1 −1 (−1)n+1 yn + 1 c5 c6 (yn + 1) c2 c−1 c4 3 2 2 2n , yn+1 yn−1 = c−1 2 c3 c5 c1 (−1)n n (−1)n n y n + c5 c6 c1 y n + c3 c4 c1 (−1)n+1 (−1)n (zn + 1) c−1 c c c z + 1 5 n 3 4 6 2n , zn+1 zn−1 = c2 c23 c−2 5 c1 (−1)n+1 n (−1)n n zn + c2 c−1 c c z n + c3 c4 c1 5 6 1 (4.28) (−1)n −1 (−1)n+1 Yn + 1 c5 c6 (Yn + 1) c3 c4 2 2n , Yn+1 Yn−1 = c2 c−2 3 c5 c1 (−1)n n (−1)n+1 n Y + c c c Yn + c2 c−1 c c n 5 6 1 3 4 1 (−1)n (−1)n −1 Zn + 1 c5 c6 (Zn + 1) c2 c3 c4 −2 2n , Zn+1 Zn−1 = c32 c−2 3 c5 c1 (−1)n+1 n (−1)n+1 n Zn + c2 c−1 c1 Zn + c2 c−1 c1 5 c6 3 c4 where c1 , c2 , . . . , c6 are conserved quantities c21 =

yn+1 zn+1 Yn+1 Zn+1 , yn zn Yn Zn

c2 = yn zn Yn Zn c−2n , 1

c23 = y2n+1 z2n+1 y2n z2n c1

,

c25 = y2n+1 Y2n+1 y2n Y2n c1

,

−(4n+1)

−(4n+1)

y2n z2n c1 , y2n+1 z2n+1 y2n Y2n c1 . c26 = y2n+1 Y2n+1 c24 =

(4.29)

If we put fn := y2n , gn := y2n+1 , we obtain −1 (gn + 1) c2 c−1 3 c4 c5 c6 g n + 1 fn+1 fn = 2n+1 , 2n+1 gn + c5 c−1 gn + c3 c−1 6 c1 4 c1 −1 −1 −1 −1 −1 2 2 4n (fn + 1) c2 c3 c4 c5 c6 fn + 1 gn gn−1 = c2 c3 c5 c1 2n (fn + c3 c4 c2n 1 ) (fn + c5 c6 c1 ) c2−1 c23 c25 c4n+2 1

(4.30)

from (4.20). These equations are nothing but the q-Painlev´e VI equation [8]. Proof By the definition of mutation (2.1), the coefficients yn,m , zn,m , Yn,m , Zn,m satisfy −1

−1 Yn−1 +1

−1

−1 Zn−1 +1

−1 yn = yn,n−1 (yn−1 + 1) zn−1 +1

yn,n−1 =

−1 Zn−2 ,

−1 zn = zn,n−1 (zn−1 + 1) yn−1 +1

zn,n−1 =

−1 Yn−2 ,

−1 Yn = Yn,n−1 (Yn−1 + 1) Zn−1 +1 −1 Yn,n−1 = zn−2 ,

Zn,n−1 =

(Zn−1 + 1),

−1

(Yn−1 + 1),

−1

−1 yn−1 +1

−1

−1 zn−1 +1

−1 Zn = Zn,n−1 (Zn−1 + 1) Yn−1 +1 −1 yn−2 ,

−1

(4.31)

−1

(zn−1 + 1),

−1

(yn−1 + 1),

where we consider the index n of the coefficients yn,m , zn,m , Yn,m , Zn,m as n ∈ Z/2Z. The equations only

14

for yn , zn , Yn , Zn we obtain, are: (yn+1 + 1)(Zn+1 + 1) −1 , −1 zn+1 + 1 Yn+1 + 1 Zn (zn+1 + 1)(Yn+1 + 1) −1 , = −1 yn+1 + 1 Zn+1 + 1 Yn

yn+2 =

(4.32)

zn+2

(4.33)

(Yn+1 + 1)(zn+1 + 1) −1 , −1 Zn+1 + 1 yn+1 + 1 zn

Yn+2 =

(Zn+1 + 1)(yn+1 + 1) −1 . −1 Yn+1 + 1 zn+1 + 1 yn

Zn+2 =

(4.34) (4.35)

We have yn+2 zn+2 Yn+2 Zn+2 yn+1 zn+1 Yn+1 Zn+1 = , yn+1 zn+1 Yn+1 Zn+1 yn zn Yn Zn yn+2 zn+2 Yn Zn = yn+1 zn+1 Yn+1 Zn+1 , yn+2 Yn+2 zn Zn = yn+1 zn+1 Yn+1 Zn+1

(4.36) (4.37) (4.38)

from (4.32)×(4.33)×(4.34)×(4.35), (4.32)×(4.33), and (4.32)×(4.34) respectively, and we obtain yn+1 zn+1 Yn+1 Zn+1 = c21 yn zn Yn Zn

(4.39)

from (4.36), where c1 is a constant. In fact, yn zn Yn Zn = c2 c2n 1

(4.40)

from yn+1 zn+1 Yn+1 Zn+1 = c21 yn zn Yn Zn , where c2 is a constant. Eliminating Yn , Zn from (4.37) and (4.40) we obtain yn+2 zn+2 = c21 yn zn . If we eliminate zn , Zn from (4.38) and (4.40) we obtain yn+2 Yn+2 = c21 yn Yn . Therefore, (−1)n n (−1)n n y n z n = c3 c4 c1 , yn Yn = c5 c6 c1 , (4.41) where c3 , c4 , c5 , c6 are constants. Eliminating zn , Yn from (4.40) and (4.41) we obtain (−1)n

yn = c−1 2 c3 c4

(−1)n

c5 c6

Zn .

(4.42)

Finally, eliminating zn , Yn , Zn from (4.41), (4.42), and (4.32) we obtain the q-Painlev´e VI equation (4.28). The equations for zn , Yn , Zn are the same as that for yn .

5

Conclusion

We have shown that cluster variables can satisfy the discrete KdV equation, the Hirota-Miwa equation, the discrete mKdV equation, and the discrete Toda equation, if we take appropriate quivers of initial seeds. We have also shown that the coefficients of certain cluster algebras satisfy the q-Painlev´e I,II,III,VI equations. These cluster algebras are obtained from a reduction of the quivers for some integrable partial difference equations. The q-Painlev´e III equation (4.20) and the q-Painlev´e VI equation (4.28) (1) are classified as type (A2 + A1 )(1) and D5 in the classification of root systems [8]. So far we have not (1) (1) obtained the q-discrete Painlev´e equations of type A4 and En from cluster algebras. In the future, we wish to clarify the relations between these equations and cluster algebras. Quivers of higher order analogue of q-Painlev´e I,II equations are obtained in [9]. To obtain quivers of higher order analogues of the q-Painlev´e III,VI equations is also a problem we wish to address in the future.

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[2] S. Fomin, “Total positivity and cluster algebras”, Proceedings of the International Congress of Mathematicians, 2, (2010), 125-145. [3] A. P. Fordy and R. J. Marsh, “Cluster mutation-periodic quivers and associated Laurent sequences”, Journal of Algebraic Combinatorics, 34, no. 1, (2011), 19-66. [4] R. Hirota, “Nonlinear Partial Difference Equations. I. A Difference Analogue of the Korteweg-de Vries Equation”, Journal of the Physical Society of Japan, 43, (1977), 1424-1433. [5] T. Miwa, “On Hirota’s difference equations”, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 58, no. 1, (1982), 9-12. [6] N. Okubo, “Discrete integrable systems and cluster algebras”, RIMS Kokyuroku Bessatsu, Research Institute for Mathematical Sciences, B41, (2013), 25-42. [7] R. Hirota, “Discrete Analogue of a Generalized Toda Equation”, JPSJ, 50, (1981), 3785-3791. [8] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlev´e equations”, Communications in Mathematical Physics, 220, (2001), 165-229. [9] A. N. W. Hone and R. Inoue, “Discrete Painlev´e equations from Y-systems”, J. Phys. A: Math. Theor., 47, 474007, (2014). [10] B. Grammaticos and A. Ramani, “The hunting for the discrete Painlev´e equations”, Reg. and Chaot. Dyn. , 5, (2000), 53-66. [11] M. D. Kruskal, K. M. Tamizhmani, B. Grammaticos and A. Ramani, “Asymmetric discrete Painlev´e equations”, Reg. Chaot. Dyn. 5, No. 3, (2000), 273-280. [12] K. Kajiwara, K. Kimura, “On a q-difference Painlev´e III equation: I. Derivation, symmetry and Riccati type solutions”, J. Nonlin. Math. Phys. , 10, (2003), 86-102.

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