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 in [YZ14] as an algebraic formalism of 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, and Wendlandt [GRW18].
Gufang Zhao is affiliated to IST Austria, Hausel group until July of 2018. Supported by the Advanced Grant Arithmetic and Physics of Higgs moduli spaces No. 320593 of the European Research Council.
Yang Y, Zhao G. The PBW theorem for affine Yangians. Transformation Groups. 2020;25:1371-1385. doi:10.1007/s00031-020-09572-6
Yang, Y., & Zhao, G. (2020). The PBW theorem for affine Yangians. Transformation Groups. Springer Nature. https://doi.org/10.1007/s00031-020-09572-6
Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” Transformation Groups. Springer Nature, 2020. https://doi.org/10.1007/s00031-020-09572-6.
Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” Transformation Groups, vol. 25. Springer Nature, pp. 1371–1385, 2020.
Yang Y, Zhao G. 2020. The PBW theorem for affine Yangians. Transformation Groups. 25, 1371–1385.
Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” Transformation Groups, vol. 25, Springer Nature, 2020, pp. 1371–85, doi:10.1007/s00031-020-09572-6.