# Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)

## TL;DR

This paper reviews recent advances in R-matrices, their geometric interpretation via symplectic geometry, their role in categorification of cluster algebras, and their impact on quantum integrable systems.

## Contribution

It synthesizes developments from different approaches, highlighting new geometric tools, categorification results, and proofs of conjectures in quantum integrable systems.

## Key findings

- Geometric approach to R-matrices using stable envelopes
- Proof of a conjecture on categorification of cluster algebras
- Resolution of several conjectures in quantum integrable systems

## Abstract

R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov have given a geometric approach to R-matrices with new tools in symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a conjecture on the categorification of cluster algebras by using R-matrices in a crucial way. Eventually, a better understanding of the action of transfer-matrices obtained from R-matrices led to the proof of several conjectures about the corresponding quantum integrable systems.