Classical limits of quantum toroidal and affine Yangian algebras

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.

v1: 10 pages; v2: 12 pages, minor corrections, some details added

References in corpus (3)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Bethe subalgebras of quantum affine gl(n) via shuffle algebras
• Homomorphisms between different quantum toroidal and affine Yangian algebras

Cited by in corpus (5)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• A slow review of the AGT correspondence
• Homomorphisms between different quantum toroidal and affine Yangian algebras
• Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)
• The formal shift operator on the Yangian double