Classical limits of quantum toroidal and affine Yangian algebras

TL;DR

This paper computes the classical limits of quantum toroidal and affine Yangian algebras of sl(n), extending previous work on gl(1), and discusses implications for algebraic structures and isomorphisms.

Contribution

It generalizes the classical limit computations from gl(1) to sl(n) for quantum toroidal and affine Yangian algebras, providing new insights and alternative proofs.

Findings

Classical limits of the algebras are explicitly computed.

New isomorphisms for n>2 are established.

Implications for algebraic structures are discussed.

Abstract

In this short note, we compute the classical limits of the quantum toroidal and the affine Yangian algebras of sl(n) by generalizing our arguments for gl(1) from arXiv:1404.5240. These results were mentioned as motivation in the recent study of toroidal algebras by [Feigin-Jimbo-Miwa-Mukhin], while they played a crucial role in the recent paper arXiv:1512.09109 of the author and Bershtein. An alternative proof of our key isomorphisms for n>2 was first established by [Varagnolo-Vasserot]. We also discuss some consequences of these results.