# Shuffle algebras associated to surfaces

## TL;DR

This paper explores the algebraic structure of Hecke correspondences on the K-theory of moduli spaces of stable sheaves on surfaces, revealing quadratic relations and connections to known algebras like Ding-Iohara-Miki.

## Contribution

It introduces quadratic relations for Hecke correspondences and compares the generated algebra with Ding-Iohara-Miki and shuffle algebras, advancing understanding of their interrelations.

## Key findings

- Derived quadratic relations between Hecke correspondences
- Established isomorphism with Ding-Iohara-Miki algebra under certain conditions
- Connected the algebra to generalized shuffle algebras

## Abstract

We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface S. We derive quadratic relations between the Hecke correspondences, and compare the algebra they generate with the Ding-Iohara-Miki algebra (at a suitable specialization of parameters), as well as with a generalized shuffle algebra.