# Genus Two Generalization of $A_1$ spherical DAHA

## TL;DR

This paper introduces a genus two generalization of the $A_1$ spherical DAHA using a system of difference operators, connecting algebraic structures with the topology of a genus two surface.

## Contribution

It constructs a new algebra generalizing $A_1$ spherical DAHA to genus two, incorporating automorphisms related to the mapping class group of a genus two surface.

## Key findings

- Automorphisms satisfy all relations of the genus two surface's mapping class group.
- The algebra generalizes $A_1$ spherical DAHA topologically.
- Provides a new framework linking algebraic and topological structures.

## Abstract

We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by $x_{ij} + x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of the closed genus two surface. Therefore we argue from topological perspective this algebra is a genus two generalization of $A_1$ spherical DAHA.