Generalized q-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

Author:

Okubo Naoto1,Suzuki Takao2

Affiliation:

1. Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1, Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa 252-5258, Japan

2. Department of Mathematics, Kindai University, 3-4-1, Kowakae, Higashi-Osaka, Osaka 577-8502, Japan

Abstract

Abstract In this article we formulate a group of birational transformations that 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.

Funder

Promotion of Science KAKENHI

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference38 articles.

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3. Drinfeld–Sokolov hierarchies of type $A$ and fourth order Painlevé systems;Fuji;Funkcial. Ekvac.,2010

4. Higher order Painlevé systems of type A. Drinfeld–Sokolov hierarchies and Fuchsian systems;Fuji;RIMS Kokyuroku Bessatsu,2012

5. Cluster algebras I: foundations;Fomin;J. Amer. Math. Soc.,2002

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