Symmetries in Connection Preserving Deformations

TL;DR

This paper explores the structure of the root lattice associated with Bäcklund transformations of certain q-Painlevé equations, revealing how it can be represented as a quotient of the lattice of connection preserving deformations and highlighting symmetries via Dynkin diagram automorphisms.

Contribution

It demonstrates that the root lattice of Bäcklund transformations can be expressed as a quotient of the lattice of connection preserving deformations, linking algebraic structures and symmetries.

Findings

Root lattice of Bäcklund transformations is a quotient of the lattice of connection preserving deformations.

Various directions in the lattice correspond to equivalent evolution equations.

Dynkin diagram automorphisms relate different evolution equations.

Abstract

We wish to show that the root lattice of B\"acklund transformations of the $q$-analogue of the third and fourth Painlev\'e equations, which is of type $(A_2+ A_1)^{(1)}$, may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we will show various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms.