# Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra

## TL;DR

This paper explores the construction of Maulik-Okounkov's R-matrix within the Ding-Iohara-Miki algebra framework, providing a boson realization and verifying its consistency, thus linking integrable models to gauge theory via algebraic structures.

## Contribution

It presents a concrete boson realization of the universal R-matrix in DIM algebra and derives MO's R-matrix conditions from this free boson oscillator expression.

## Key findings

- Boson realization of the universal R-matrix in DIM algebra
- Derivation of MO's R-matrix conditions from oscillator expression
- Consistency checks confirming the validity of the boson realization

## Abstract

The integrability of 4d $\mathcal{N}=2$ gauge theories has been explored in various contexts, for example the Seiberg-Witten curve and its quantization. Recently, Maulik and Okounkov proposed that an integrable lattice model is associated with the gauge theory, through an R-matrix, to which we refer as MO's R-matrix in this paper, constructed in the instanton moduli space. In this paper, we study the R-matrix using the Ding-Iohara-Miki (DIM) algebra. We provide a concrete boson realization of the universal R-matrix in DIM and show that the defining conditions for MO's R-matrix can be derived from this free boson oscillator expression. Several consistency checks for the oscillator expression are also performed.