The PBW theorem for the affine Yangians

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.