Quantum toroidal and shuffle algebras

TL;DR

This paper establishes an isomorphism between the quantum toroidal algebra of gl_n and the double shuffle algebra, enabling new insights into the structure and factorization of the universal R-matrix.

Contribution

It demonstrates the isomorphism between quantum toroidal and shuffle algebras, providing a new approach to analyze the universal R-matrix.

Findings

Proves the isomorphism between quantum toroidal and shuffle algebras.

Derives a factorization formula for the universal R-matrix.

Introduces a shuffle algebra perspective for quantum toroidal algebras.

Abstract

In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra.