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Rei Inoue

Publications and source records attributed to Rei Inoue.

At least 19 recordsLinked to original sources
arXiv2025

Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the $q$-Weyl algebra. The new $K$-operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.

math.QA↗
arXiv2025

Quantized six-vertex model on a torus

We study the integrability of the quantized six-vertex model with four parameters on a torus. It is a three-dimensional integrable lattice model in which a layer transfer matrix, depending on two spectral parameters associated with the homology cycles of the torus, can be defined not only on the square lattice but also on more general graphs. For a class of graphs that we call admissible, we establish the commutativity of the layer transfer matrices by introducing four types of tetrahedron equations and two types of inversion relations. Expanding in the spectral parameters yields a family of commuting quantum Hamiltonians. The quantized six-vertex model can also be reformulated in terms of (quantized) dimer models, and encompasses known integrable systems as special cases, including the free parafermion model and the relativistic Toda chain.

nlin.SI↗
arXiv2024

Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type $A$ Weyl groups and the implementation of quantum $Y$-variables through the $q$-Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) $R$-matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D $R$-matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.

math.QA↗
arXiv2023

Quantum cluster algebras and 3D integrability: Tetrahedron and 3D reflection equations

We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock-Goncharov quivers associated with the longest elements of the Weyl groups of type $A$ and $C$, and investigate the cluster transformations corresponding to changing a reduced expression into a `most distant' one. By devising a new realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum $y$-variables. Explicit formulas of their matrix elements are also derived for some typical representations.

math.QA↗
arXiv2023

Tetrahedron equation and quantum cluster algebras

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms $q$-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's $R$ matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when $q$ is specialized to a root of unity.

math.QA↗
arXiv2022

Invariants of Weyl group action and $q$-characters of quantum affine algebras

Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here $\mathcal{Y}_m$ is the rational function field on $cm\ell$ commuting variables, where $c \in \{ 1, 2, 3 \}$ depends on $\mathfrak{g}$. This was motivated by the $q$-character map $\chi_q$ of the category of finite dimensional representations of quantum affine algebra $U_q(\hat{\mathfrak{g}})$. We showed in [I21] that when $q$ is a root of unity, $\mathrm{Im} \chi_q$ is a subring of the $W$-invariant subfield $\mathcal{Y}_m^W$ of $\mathcal{Y}_m$. In this paper, we give more detailed study on $\mathcal{Y}_m^W$; for each reflection $r_i \in W$ associated to the $i$th simple root, we describe the $r_i$-invariant subfield $\mathcal{Y}_m^{r_i}$ of $\mathcal{Y}_m$.

math.RT↗
arXiv2020

Cluster realization of Weyl groups and $q$-characters of quantum affine algebras

We consider an infinite quiver $Q(\mathfrak{g})$ and a family of periodic quivers $Q_m(\mathfrak{g})$ for a finite dimensional simple Lie algebra $\mathfrak{g}$ and $m \in \mathbb{Z}_{>1}$. The quiver $Q(\mathfrak{g})$ is essentially same as what introduced by Hernandez and Leclerc for the quantum affine algebra. We construct the Weyl group $W(\mathfrak{g})$ as a subgroup of the cluster modular group for $Q_m(\mathfrak{g})$, in a similar way as what studied by the author, Ishibashi and Oya, and study its applications to the $q$-characters of quantum non-twisted affine algebras $U_q(\hat{\mathfrak{g}})$ introduced by Frenkel and Reshetikhin, and to the lattice $\mathfrak{g}$-Toda field theory. In particular, when $q$ is a root of unity, we prove that the $q$-character is invariant under the Weyl group action. We also show that the $A$-variables for $Q(\mathfrak{g})$ correspond to the $\tau$-function for the lattice $\mathfrak{g}$-Toda field equation.

math.RT↗
arXiv2019

Cluster realizations of Weyl groups and higher Teichm\"uller theory

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a family of weighted quivers $Q_m(\mathfrak{g})$ ($m \geq 2$) whose cluster modular group $\Gamma_{Q_m(\mathfrak{g})}$ contains the Weyl group $W(\mathfrak{g})$ as a subgroup. We compute explicit formulae for the corresponding cluster $\mathcal{A}$- and $\mathcal{X}$-transformations. As a result, we obtain green sequences and the cluster Donaldson-Thomas transformation for $Q_m(\mathfrak{g})$ in a systematic way when $\mathfrak{g}$ is of finite type. Moreover if $\mathfrak{g}$ is of classical finite type with the Coxeter number $h$, the quiver $Q_{kh}(\mathfrak{g})$ ($k \geq 1$) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichm\"uller space of a once-punctured disk with $2k$ marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichm\"uller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS18] from the geometrical viewpoint.

math.RT↗
arXiv2017

Soliton cellular automata associated with infinite reduced words

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

nlin.SI↗
arXiv2016

On the cluster nature and quantization of geometric $R$-matrices

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

math.QA↗
arXiv2015

Toric networks, geometric $R$-matrices and generalized discrete Toda lattices

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.

math.AG↗
arXiv2014

Discrete Painlevé equations from Y-systems

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

math-ph↗
arXiv2014

Braiding Operator via Quantum Cluster Algebra

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-matrix up to a simple gauge-transformation.

math.QA↗
arXiv2013

Braids, Complex Volume, and Cluster Algebra

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

math.GT↗
arXiv2012

Tropical curves and integrable piecewise linear maps

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.

math-ph↗
arXiv2011

Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the 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 article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.

math-ph↗