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## Discrete integrable systems and Poisson algebras from cluster maps (2014)

Citations: | 5 - 2 self |

### Citations

270 |
A simple model of the integrable Hamiltonian equation
- Magri
- 1978
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Citation Context ...oisson bracket after scaling Jj → µJj for arbitrary µ, the brackets {, }0 and {, }2 are compatible, so define a bi-Hamiltonian structure. This means that one can use the standard bi-Hamiltonian chain =-=[26]-=-, defining a sequence of functions Ij which satisfy {Ij , Ik}0 = 0 = {Ij , Ik}2, for all j, k, where the sequence starts from I0 = ∑ j Jj , the Casimir of the bracket {, }0 given by (4.20), and finish... |

238 |
Cluster algebras
- Fomin, Zelevinsky
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Citation Context ...ywords : Integrable maps, Poisson algebra, cluster algebra, algebraic entropy, tropical, monodromy. 1 Introduction Cluster algebras were first developed by Fomin and Zelevinsky more than a decade ago =-=[10]-=-. Their structure arises in diverse parts of mathematics and theoretical physics, including Lie theory, quantum algebras, Teichmüller theory, discrete integrable systems and T- and Y-systems. One of ... |

235 |
Integral Matrices
- Newman
- 1972
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Citation Context ....19) is trivial for π = id, but there are other non-trivial choices of π, corresponding to different integer bases for imB. The following classical theorem (which is a special case of Theorem IV.1 in =-=[29]-=-) provides a canonical choice of π, and via Lemma 2.7 gives Darboux coordinates for the presymplectic form ω. The proof in [29] is constructive. Theorem 2.8. If B is a skew-symmetric matrix of rank 2K... |

81 |
Dressing chains and the spectral theory of the Schro ¨ dinger operator
- Veselov, Shabat
- 1993
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Citation Context ...p+1 − 1). (4.16) Remark 4.7. An alternative formula for K is given by a link with the dressing chain: K = p∏ j=1 ( 1− ∂ 2 ∂Jj∂Jj+1 ) p∏ k=1 Jk. When p is odd, this formula follows from the results in =-=[38]-=-, by setting βi → 0 and gi → Ji. 4.3.1 Link with the Poisson structure In the case N = 2m, we can use the monodromy matrix to build the Poisson bracket for the functions Jn, Kn. Staying within the con... |

69 |
The Laurent Phenomenon
- Fomin, Zelevinsky
- 2002
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Citation Context ...s (2.4), the key to calculating their entropy is the Laurent phenomenon, which leads to an exact recursion for the degrees of the denominators. The Laurent property for the associated cluster algebra =-=[11]-=- implies that the iterates have the factorized form xn = Nn(x) Mn(x) , with Nn ∈ Z[x] = Z[x1, . . . , xN ], Mn = N∏ j=1 x d(j)n j , where the polynomials Nn are not divisible by xj for 1 ≤ j ≤ N , and... |

60 | Cluster algebras and Weil–Petersson forms
- Gekhtman, Shapiro, et al.
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Citation Context ...rix B corresponding to the quiver, which not only defines the exponents appearing in (1.3), but also produces a presymplectic form which is invariant under the map; this is the two-form introduced in =-=[18]-=-. When detB 6= 0, the form is symplectic, so the map automatically has a nondegenerate Poisson bracket. The main result of section 2 is Theorem 2.6, which states that (even if detB = 0) it is always p... |

53 | Algebraic entropy
- Bellon, Viallet
- 1999
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Citation Context ...ovides us with the appropriate setting in which to consider Liouville integrability in the rest of the paper. In section 3 we consider the recurrences (1.3) in the light of the algebraic entropy test =-=[2]-=-. We give details of a series of conjectures which show that the algebraic entropy can be determined explicitly from the tropical version of (1.3), expressed in terms of the max-plus algebra. From the... |

50 | The Pentagram map: a discrete integrable system. arXiv:0810.5605v2 [math.DS] 14
- Ovsienko, Schwartz, et al.
- 2009
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Citation Context ...) follows by symmetry, from considering the left kernel of ∆n. Remark 5.2. The four-term linear relations (5.4) and (5.5), together with det Ψ̃n = 1, should be compared with those of the pentagrammap =-=[31]-=-, but there the coefficients of the second and third terms are independent. Remark 5.3. When q = 1 the coefficient Kn has period 1, so Kn+1 = Kn = K for all n, and the recurrence (5.5) is just the con... |

47 |
Condensation of determinants
- Dodgson
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Citation Context ...plies that the iterates of (4.1) form an infinite frieze. 15 Upon forming the matrix Ψ̃n = xn xn+q xn+2qxn+p xn+N xn+N+q xn+2p xn+N+p xn+2N , (4.3) one can use the Dodgson condensation method =-=[6]-=- to expand the 3 × 3 determinant in terms of its 2 × 2 minors, as det Ψ̃n = 1 xn+N (detΨn detΨn+N − detΨn+q detΨn+p) = 0, by (4.2). By considering the right and left kernels of Ψ̃n, we are led to the ... |

44 | Skew-symmetric cluster algebras of finite mutation type
- Felikson, Shapiro, et al.
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Citation Context ...to consider only the case where p, q are coprime, which we will assume from now on. The cluster algebras generated by affine A-type Dynkin quivers arise from surfaces, and are of finite mutation type =-=[8]-=-, meaning that only a finite number of distinct quivers is obtained under sequences of mutations (2.1). However, by the classification result in [10], these cluster algebras are not themselves finite:... |

29 |
Dimers and cluster integrable systems
- Goncharov, Kenyon
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Citation Context ...tes of the families (ii) and (iii)). Recently, Goncharov and Kenyon have found Somos recurrences arising as discrete symmetries of classical integrable systems associated with dimer models on a torus =-=[19]-=-. A further connection with relativistic analogues of the Toda lattice appeared in [7]. 42 6.1 Reductions of the Hirota-Miwa equation The relations (6.1) all arise by reduction of the Hirota-Miwa (dis... |

28 |
Cluster ensembles, quantization and the
- Fock, Goncharov
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Citation Context ...xj ∧ dxk, (2.10) which is just the constant skew-form ω = ∑ j<k bjk dzj ∧ dzk, written in the logarithmic coordinates zj = log xj , so it is evidently closed, but may be degenerate. In [18] (see also =-=[9]-=-) it was shown that for a cluster algebra defined by a skew-symmetric integer matrix B, this two-form is compatible with cluster transformations, in the sense that under a mutation map µi : x 7→ x̃, i... |

26 |
The strange and surprising saga of the Somos sequences
- Gale
- 1991
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Citation Context ...s, Teichmüller theory, discrete integrable systems and T- and Y-systems. One of the original motivations for cluster algebras came from a series of observations made by Michael Somos and others (see =-=[16]-=-), concerning nonlinear recurrence relations of the form xn+N xn = F (xn+1, . . . , xn+N−1), (1.1) where F is a polynomial in N − 1 variables. The original observation of Somos was that certain choice... |

22 | Cluster Mutation-Periodic Quivers and Associated Laurent Sequences
- Fordy, Marsh
- 2009
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Citation Context ... bracket, as well as having sufficiently many first integrals that commute with respect to this bracket. A quiver is a graph consisting of a number of nodes together with arrows between the nodes. In =-=[13]-=-, Fordy and Marsh showed how recurrences of the form (1.3) are produced from sequences of mutations in cluster algebras defined by quivers with a special periodicity property with respect to mutations... |

17 |
Cluster algebras and Poisson geometry, Mosc
- Gekhtman, Shapiro, et al.
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Citation Context ... . . . , xN ) to (x2, . . . , xN , x1). Due to the periodicity requirement on B, we have ρ−1 · µ1 (B) = B, so the action of ϕ on this matrix is trivial. 2.2 The Gekhtman-Shapiro-Vainshtein bracket In =-=[17]-=- it was shown that very general cluster algebras admit a linear space of Poisson brackets of log-canonical type, compatible with the cluster maps generated by mutations, and having the form {xj , xk} ... |

13 | 2-Frieze patterns and the cluster structure of the space of polygons
- Morier-Genoud, Ovsienko, et al.
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Citation Context ...s with periodic coefficients By analogy with (4.2), the recurrence (5.1) can be written in the form det Ψn = ∣∣∣∣ xn xn+qxn+p xn+N ∣∣∣∣ = xn+m. (5.3) The above identity is the relation for a 2-frieze =-=[28]-=-, and it implies that the iterates of (5.1) can be placed in the form an infinite 2-frieze. Using Dodgson condensation once again to condense a 3 × 3 determinant, we have det Ψ̃n = (xn+mxn+N+m − xn+m+... |

11 |
Some periodic and non-periodic recursions,”
- Csornyei, Laczkovich
- 2001
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Citation Context ...opriate choice of basis for imB gives the reduction from (3.8) to (3.9), by setting Yn = dn+4− dn+3− 2dn+2− dn+1 + dn. It can be shown directly that all the orbits of (3.9) are periodic with period 9 =-=[5]-=-, and hence in this case the degrees dn satisfy a linear recurrence of order 13, that is (S9 − 1)Yn = (S9 − 1)(S4 −S3 − 2S2 −S + 1)dn = 0. From the periodicity of Yn it is clear that dn+4 − dn+3 − 2dn... |

10 | Algebraic entropy of birational maps with invariant curves
- Bellon
- 1999
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Citation Context ... zero algebraic entropy. In an algebrogeometric setting, there are plausible arguments which indicate that zero entropy should be a necessary condition for integrability in the Liouville-Arnold sense =-=[3]-=-. In the latter setting, each iteration of the map corresponds to a translation on an Abelian variety (the level set of the first integrals), and the degree is a logarithmic height function, which nec... |

10 | Linear recurrence relations for cluster variables of affine quivers
- Keller, Scherotzke
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Citation Context ...es in this case satisfy linear recurrence relations with constant coefficients. (This was subsequently shown for the general case of cluster algebras associated with affine Dynkin quivers, in [1] and =-=[23]-=-.) Here we give a new proof of these linear recurrences, which relies on additional linear relations with periodic coefficients, and their associated monodromy matrices. These periodic quantities are ... |

10 | Recurrence relations for elliptic sequences: every Somos 4 is a Somosk
- Poorten, Swart
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Citation Context ... [20]). However, the quantity Ĩ is not defined on the (y1, y2) plane, where there is only one first integral (as required for the Liouville-Arnold theorem). 6.2 Bilinear relations of higher order In =-=[32]-=-, Swart and van der Poorten proved that sequences generated by Somos-4 recurrences also satisfy quadratic (Somos-type) relations of order k, for all k ≥ 4. They also noted that for Somos-5 sequences, ... |

10 |
Periodicity of Somos sequences
- Robinson
- 1992
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Citation Context ...f the 8-dimensional symplectic map (5.36). 6 Integrable maps from Somos sequences The quadratic recurrences (3.15) (case (iv) of Theorem 3.12), are referred to as three-term Gale-Robinson recurrences =-=[16, 34]-=-. We mow consider the slightly more general case where these recurrences have coefficients: xn+N xn = αxn+N−p xn+p + β xn+N−q xn+q. (6.1) These can be included by adding extra nodes to the quiver for ... |

10 |
A geometric approach to singularity confinement and algebraic entropy
- Takenawa
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Citation Context ...ical as the dimension increases, and provides no proof that the linear relation, with its corresponding entropy value, is correct. In dimension two, exact results are possible via intersection theory =-=[35, 36]-=-. In the rest of this section we seek to isolate those recurrences with E = 0, by finding a condition on the exponents which should be necessary and sufficient for E > 0. The main conjecture is the fo... |

9 | Sigma function solution of the initial value problem for Somos 5 sequences
- Hone
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Citation Context ...ntegral, also of degree 5 in terms of the cluster variables, which can be written as Ĩ = f1f2f3 + α̃ ( 1 f1 + 1 f2 + 1 f3 ) + β̃ f1f2f3 with fj = xjxj+2 x2j+1 for j = 1, 2, 3 (see Proposition 2.3 in =-=[20]-=-). However, the quantity Ĩ is not defined on the (y1, y2) plane, where there is only one first integral (as required for the Liouville-Arnold theorem). 6.2 Bilinear relations of higher order In [32],... |

9 |
A Ultradiscrete QRT maps and tropical elliptic curves
- Nobe
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Citation Context .... . . , N , and the degrees d (j) n for 2 ≤ j ≤ N have the same initial data but shifted along by an appropriate number of steps. 11 Remark 3.3. The recurrence (3.3) is the tropical (or ultradiscrete =-=[30]-=-) analogue of the original nonlinear recurrence (2.4), in terms of the max-plus algebra. It is a special case of the recursion for the denominator vectors in a general cluster algebra, which is stated... |

7 |
Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures
- Byrnes, Haggar, et al.
- 1999
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Citation Context .... The existence of N − 1 independent first integrals Ij , together with the volume form Ω in (2.17), means that the map ϕ is Liouville integrable in a rather elementary sense. By applying a method in =-=[4]-=-, one can pick, say, the first N − 2 integrals and obtain a Poisson bivector field Ĵ which is invariant (or anti-invariant, for odd N), namely Ĵ = Ω̂ydI1y . . .ydIN−2, where the N -vector field Ω̂ i... |

7 |
Quantum Integrable Models and Discrete
- Krichever, Lipan, et al.
- 1997
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Citation Context ...Laurent property for three-term Somos (or Gale-Robinson) recurrences of the form (6.1) then follows by the reduction (6.3). The Hirota-Miwa equation (6.2) has a scalar Lax pair (see equation (3.8) in =-=[24]-=-, for instance): it is the compatibility condition for the linear system given by T−1,3 ψ1,2 + T ψ2,3 = T2,3 ψ, T ψ−1,2 + T−1,3 ψ2,−3 = T−1,2 ψ, (6.4) in terms of the scalar function ψ = ψ(n1, n2, n3)... |

6 | Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
- Fordy
- 2010
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Citation Context ...n Theorem 4.8. The function K, defined by (4.12) is the Casimir of the Poisson bracket of Lemma 4.9: (P(2) +P(0))∇K = 0. 4.4 Poisson brackets and Liouville integrability for P (1) 2m It was proved in =-=[14]-=- that the linearisable maps coming from the primitives P (1) N (the Ã1,N−1 Dynkin quivers) are Liouville integrable when N is even. We give the proof here, since it is the basis for understanding the... |

6 | Symplectic Maps from Cluster Algebras
- Fordy, Hone
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Citation Context ...s family, such as making reductions of the Hirota-Miwa equation and its Lax pair and by deriving higher bilinear relations with constant coefficients. Some of our results were announced previously in =-=[15]-=-. 2 Symplectic maps from cluster recurrences Given a recurrence, a major problem is to find an appropriate symplectic or Poisson structure which is invariant under the action of the corresponding fini... |

3 |
Algebraic dynamics and algebraic entropy
- Viallet
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Citation Context ...same symplectic form as in (2.26). The corresponding symplectic map is ϕ̂ : ( y1 y2 ) 7−→ ( y2 (y22 + 1)/(y1y2) ) , (2.28) whose singularity pattern under successive blowups was considered by Viallet =-=[36]-=-. The map (2.28) has positive algebraic entropy, indicating nonintegrability. (See Example 3.7 in the next section.) However, one can take a different basis, corresponding to case (b) of Lemma 2.9, gi... |

2 |
Analytic solutions and integrability for bilinear recurrences of order six
- Hone
- 2010
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Citation Context ... this subsection we explain how to obtain first integrals for Somos-7 recurrences using associated quadratic (bilinear) relations of higher order. The analogous results for Somos-6 recurrences are in =-=[22]-=-. To present the results concisely, it is convenient to consider the most general form of a Somos-7 recurrence, which is the four-term Gale-Robinson relation xn+7 xn = α xn+6 xn+1 + βxn+5 xn+2 + γ xn+... |

2 |
Integrable mappings and soliton equations, Physics Letters A 126
- Quispel, Roberts, et al.
- 1988
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Citation Context ...es the rational map π : C4 → C2 from the xj to the yj . Upon computing the pullback of ϕ on these monomials, one has the map ϕ̂ : ( y1 y2 ) 7−→ ( y2 (y2 + 1)/(y1y 2 2) ) , (2.25) which is of QRT type =-=[33]-=-, and preserves the symplectic form ω̂ = 1 y1y2 dy2 ∧ dy1 (2.26) where ω = π∗ω̂, with ω given by the formula (2.16) for c = 2. Note that, in the preceding example, the chosen basis is such that every ... |

1 |
Completely Integrable Symplectic
- Maeda
- 1987
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Citation Context ...s to identify which recurrences of the form (1.3) can be regarded as finite-dimensional discrete integrable systems, in the sense of the standard Liouville-Arnold definition of integrability for maps =-=[25, 37]-=-. The latter requires that a map should preserve a Poisson bracket, as well as having sufficiently many first integrals that commute with respect to this bracket. A quiver is a graph consisting of a n... |

1 |
Poisson reduction of the space of polygons
- Marshall
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Citation Context ...gram map. However, a Dirac reduction of this bracket to the case of (5.4) or (5.5) gives only the trivial bracket. A general approach to Poisson structures related to twisted polygons is described in =-=[27]-=-, which should shed some light on the situation here. For want of more general statements, we illustrate the foregoing discussion with several examples of the integrable systems that arise in this cas... |