arXiv subjects

Naoto Okubo

Publications and source records attributed to Naoto Okubo.

6 recordsLinked to original sources

arXiv2026

Co-primeness preserving higher dimensional extension of $q$-discrete Painlevé I, II equations

We construct the $q$-discrete Painlevé I and II equations and their higher order analogues by virtue of periodic cluster algebras. Using particular $k \times k$ exchange matrices, we show that the cluster algebras corresponding to $k=4$ and $5$ give the $q$-discrete Painlevé I and II equations respectively. For $k \ge 6$, we obtain higher-order discrete equations that satisfy an integrability criterion, namely, the co-primeness property.

math-ph↗

arXiv2025

Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras

A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to derive a tropical, i.e. subtraction-free birational, representation of Weyl groups from cluster algebras. Our results provide an extensive class of Weyl group actions, including previously known examples with algebro-geometric background, and hence are relevant to the q-Painleve equations and their higher-order extensions. Key ingredients of the argument are the combinatorial aspects of the reflection associated with a cycle subgraph in the quiver. We also study symplectic structures of the discrete dynamical systems thus obtained. The normal form of a skew-symmetric integer matrix allows us to choose Darboux coordinates while preserving the birationality.

nlin.SI↗

arXiv2020

Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra

In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo-Sakai's $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again; the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.

math.QA↗

arXiv2018

Toda type equations over multi-dimensional lattices

We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.

math-ph↗

arXiv2016

Laurent phenomenon algebras and the discrete BKP equation

We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a generalization of mutation-period property. We show that a reduction of a seed corresponds to a reduction of a difference equation.

math-ph↗

arXiv2015

Bilinear equations and $q$-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras

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 q-discrete Painlevé 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.

math-ph↗