井上 玲

We consider a family of cellular automata $\Phi(n,k)$ associated with<br /> infinite reduced elements on the affine symmetric group $\hat S_n$, which is a<br /> tropicalization of the rational maps introduced by two of the authors. We study<br /> the soliton solutions for $\Phi(n,k)$ and explore a `duality&#039; with the<br /> $\mathfrak{sl}_n$-box-ball system.

We define cluster $R$-matrices as sequences of mutations in triangular grid<br /> quivers on a cylinder, and show that the affine geometric $R$-matrix of<br /> symmetric power representations for the quantum affine algebra<br /> $U_q^\prime(\hat{\mathfrak{sl } }_n)$ can be obtained from our cluster<br /> $R$-matrix. A quantization of the affine geometric $R$-matrix is defined,<br /> compatible with the cluster structure. We construct invariants of the quantum<br /> affine geometric $R$-matrix as quantum loop symmetric functions.

We use the combinatorics of toric networks and the double affine geometric R-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the R-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

We try to give a cluster-algebraic interpretation of the complex volume of knots. We construct the R-operator from cluster mutations, and show that it can be regarded as a hyperbolic octahedron. The cluster variables are interpreted as the edge parameters used by Zickert for computing complex volume.

We apply a reduction to the Beauville systems to obtain a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism. (C) 2014 Elsevier B.V. All rights reserved.

We consider T-systems and Y-systems arising from cluster mutations applied to<br /> quivers that have the property of being periodic under a sequence of mutations.<br /> The corresponding nonlinear recurrences for cluster variables (coefficient-free<br /> T-systems) were described in the work of Fordy and Marsh, who completely<br /> classified all such quivers in the case of period 1, and characterized them in<br /> terms of the skew-symmetric exchange matrix B that defines the quiver. A<br /> broader notion of periodicity in general cluster algebras was introduced by<br /> Nakanishi, who also described the corresponding Y-systems, and T-systems with<br /> coefficients.<br /> A result of Fomin and Zelevinsky says that the coefficient-free T-system<br /> provides a solution of the Y-system. In this paper, we show that in general<br /> there is a discrepancy between these two systems, in the sense that the<br /> solution of the former does not correspond to the general solution of the<br /> latter. This discrepancy is removed by introducing additional non-autonomous<br /> coefficients into the T-system. In particular, we focus on the period 1 case<br /> and show that, when the exchange matrix B is degenerate, discrete Painlev\&#039;e<br /> equations can arise from this construction.

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type B-r at any level. We also prove the dilogarithm identities for the Y-systems of type B-r at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Using this new method, we also give an alternative and simplified proof of the periodicities of the T-systems and Y-systems associated with pairs of simply laced Dynkin diagrams.

We prove the periodicities of the restricted T-systems and Y-systems associated with the quantum affine algebra of type C-r, F-4, and G(2) at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.

The box-ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.

We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011), and on some new results obtained afterward. After a brief review on tropical curve theory, we study the spectral curves and the isolevel sets of the tropical periodic Toda lattice and the periodic Box-ball system.

We study the generalized ultradiscrete periodic Toda lattice J(M, N) which has tropical spectral curve. We introduce a tropical analogue of Fay's trisecant identity, and apply it to construct a general solution to J(M, N).

We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.

We study the singular isolevel manifold M-g (0) of the genus g Mumford system associated to the spectral curve y(2) = x(2g+1). We show that M-g (0) is stratified by g + 1 open subvarieties of additive algebraic groups of dimension 0, 1, .... , g, and we give an explicit description of M-g(0) in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus g Mumford system, which is closely related to rational solutions of the KdV hierarchy. (C) 2009 Wiley Periodicals, Inc.

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra, associated with a complex simple Lie algebra. The unrestricted T-system admits a. reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems. which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems, Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D and B (level 2).

We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function.

We prove that the general isolevel set of the ultra-discrete periodic Toda<br /> lattice is isomorphic to the tropical Jacobian associated with the tropical<br /> spectral curve. This result implies that the theta function solution obtained<br /> in the authors&#039; previous paper is the complete solution. We also propose a<br /> method to solve the initial value problem.

We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class on the tropical hyperelliptic curve. We also provide some conjectures related to the divisor class and the Jacobian. Finally, we apply our method to the periodic box and ball system and clarify the algebro-geometrical meaning of the real torus introduced for its initial value problem.

Beauville [A. Beauville, Jacobiennes des courbes spectrales et systemes hamiltoniens completement integrables, Acta. Math. 164 (1990) 211-235] introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system [D. Mumford, Tata Lectures on Theta II, Birkhauser, 1984]. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the intersection of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke [P. Vanhaecke, Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z. 211 (1992) 265-313; P. Vanhaecke, Integrable systems in the realm of algebraic geometry, in: Lecture Notes in Mathematics, vol. 1638, 2001]. (c) 2006 Elsevier B.V. All rights reserved.

We study Beauville's completely integrable system and its variant from a viewpoint of multi-Hamiltonian structures. We also relate our result to the previously known Poisson structures on the Mumford system and the even Mumford system.

The purpose of this paper is twofold. The first is to apply the method introduced in the works of Nakayashiki and Smirnov [11, 12] on the Mumford system to its variants. The other is to establish a relation between the Mumford system and the isospectral limit Q(g)((I)) and Q((II)) of the Noumi-Yamada system [15]. As a consequence, we prove the algebraically completely integrability of the systems Q(g)((I)) and Q(g)((II)), and get explicit descriptions of their solutions.

An L operator is presented related to an infinite dimensional limit of the fusion R matrices for Uq(An-1 (1)) and U q(Dn (1)). It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated.

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes M-F of polynomial matrices. Let X be the algebraic curve given by the common characteristic equation for M-F. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of X. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As an application, we discuss the algebraic complete integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.

Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka-Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi variety, and make use of it to discuss the solvability of the model. (C) 2003 American Institute of Physics.

The lattice Toda field theory for finite-dimensional simple Lie algebras is studied. We show that the Poisson structure for the lattice Toda fields is closely related to that for the q-deformed W algebra. By making use of this relationship, we construct the lattice W algebra. We discuss the cases of B-2 and C-2 in detail, and associate them with the continuous theory.

We study an integrable cellular automaton which is called the box-ball system (BBS). The BBS can be derived directly from the integrable differential-difference equation by either ultradiscretization or crystallization. We clarify the integrable structure and the hidden symmetry of the BBS.

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra U′q(A(1)M) is introduced. It is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U′q(A(1)M-1). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev-Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter. © 2001 American Institute of Physics.

The discretization of the Toda field theory for any finite-dimensional simple Lie algebras is studied. We define the Poisson relation among Toda fields on the lattice, and clarify the Hamiltonian structure of the lattice Toda field equation. We transform the Toda field equation into the functional equation of the tau-function, and discuss the correspondence with the T-system. (C) 2000 Elsevier Science B.V. All rights reserved.

We introduce the supersymmetric extension of the box-ball system. By use of the isomorphism of the crystal base for the super Lie algebra, we define the time evolution operator, and give the evolution equation explicitly. We also construct the soliton solutions.

We study the soliton cellular automaton (SCA) in (2+1)-dimensions from the viewpoint of the integrable vertex model. As in our previous paper, we relate the SCA, the so-called box-ball system, to an integrable vertex model associated with the Bogoyavlensky lattice. We extend this framework and introduce the (2 + 1)-dimensional SCA, which can be interpreted as the ultradiscretization of the 2D Toda equation. We also construct the N-soliton solutions for this system.

We introduce a vertex model in two-dimension, which is associated with the Bogoyavlensky lattice. We show that in a crystallized limit (q → 0) we have a unique configuration, and that it coincides with an evolution of the soliton cellular automata which is a generalization of the system introduced by Takahashi and Satsuma.

We construct the quantum integrable dynamical system on a (2 + 1)-dimensional lattice. We consider the q-commuting operators on 2-D lattice as a discretization of the current algebra, and two evolution operators are explicitly defined in terms of the quantum dilogarithm function.

We study the Hamiltonian structure of the Bogoyavlensky lattice, which is an integrable differential-difference equation and is a generalization of the Volterra model. We construct the lattice W algebras by use of the dynamical variables of the Bogoyavlensky lattice, and also show the bi-Hamiltonian structure thereof. The quantization of the lattice W algebras is briefly discussed.

Studied are the integrable difference-difference equations for the lattice W currents. which can be regarded as the full-discretized (reduced) KP equations. We find explicit forms of the time-discretization for the lowest equations of the hierarchy by making use of the relation with the Bogoyavlensky lattice.

We investigate the integrable structure of the quantum Volterra model, which is a discretization of the Korteweg-de Vries equation. Based on Sklyanin's method we derive the Baxter equation, and then clarify a role of the "fundamental transfer matrix". We show that the fundamental transfer matrix also generates the integrals of motion for a lattice analogue of the sine-Gordon system.

The Bogoyavlensky lattice with discrete time is studied. We introduce an integrable integrodifference equation, which is obtained by taking an interaction range to infinity in the Bogoyavlensky lattice. Its integrability is shown by constructing the Lax form. Also studied is a relationship with the fully discrete W currents.

We study the dynamical system on the space-time lattice based on solutions of the Yang-Baxter equation. We construct the quantum integrable difference-difference equations associated with the lattice W-N algebra. The classical lattice system and its time continuous limit are also discussed.

An extension of the Lotka-Volterra model, which we call the hungry-Volterra model or the Bogoyavlensky lattice, is studied. We show that the model is integrable in both classical and quantum levels by use of the local Lax matrix. The relationship between the Lotka-Volterra model and some integrable systems is briefly discussed.

We reveal the hidden algebraic symmetry of the Bogoyavlensky lattice (the extended Volterra model). Shown is the correspondence between the lattice W algebra and the Bogoyavlensky lattice. Soliton solutions are also constructed in terms of the vertex operators.

Several aspects of the lattice WN algebra are studied. Motivated by the fact that the Lotka-Volterra model can be written in terms of a current of the lattice Virasoro algebra (the Faddeev-Takhtajan-Volkov algebra), integrable dynamical models on the lattice have been formulated as a model associated with the lattice W3 algebra.

Studied is the discrete analogue of the classical W-N algebra. We derive a new representation of the exchange algebra. We discuss in detail a construction of the currents of the lattice W-3 algebra.

A new hierarchy of integrable nonlinear differential-difference equations is reported. Through an extension of the Miura transformation, such a series of discrete models are obtained from the hungry Volterra model. In addition, the relation to the lattice W-algebra is discussed.