On the geometry of a 4-dimensional extension of a $q$-Painlev\'e I equation with symmetry type $A_1^{(1)}$

TL;DR

This paper explores the geometric structure of a four-dimensional extension of a $q$-Painlevé I equation, revealing its integrability and symmetries through algebraic geometry and group actions.

Contribution

It introduces a new 4D integrable system extending $q$-Painlevé I, constructed via blow-ups and symmetry group actions, and analyzes its geometric and algebraic properties.

Findings

The system is integrable with conserved quantities.

The variety admits an extended affine Weyl group action.

Two 4D analogues of $q$-Painlevé equations are constructed.

Abstract

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlev\'e I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlev\'e equations, one of which is a deautonomisation of the original autonomous integrable map.