The PBW theorem for the affine Yangians

TL;DR

This paper proves the PBW theorem for affine Yangians associated with untwisted symmetric affine Kac-Moody Lie algebras by establishing an isomorphism with the Drinfeld double of a shuffle algebra, providing a new algebraic perspective.

Contribution

It establishes the PBW theorem for this class of affine Yangians via an isomorphism with a shuffle algebra's Drinfeld double, offering a novel proof approach.

Findings

Affine Yangian is isomorphic to the Drinfeld double of a shuffle algebra.

PBW theorem holds for the class of affine Yangians considered.

Provides an algebraic formalism connecting cohomological Hall algebras and affine Yangians.

Abstract

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.