THE PBW THEOREM FOR AFFINE YANGIANS
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Springer Science+Business Media New York (2020)
THE PBW THEOREM FOR AFFINE YANGIANS YAPING YANG
GUFANG ZHAO∗
School of Mathematics and Statistics The University of Melbourne Parkville VIC 3010, Australia
School of Mathematics and Statistics The University of Melbourne Parkville VIC 3010, Australia and Institute of Science and Technology Austria Am Campus, 1 Klosterneuburg 3400, Austria
[email protected]
[email protected]
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 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].
Introduction For a symmetrizable Kac–Moody Lie algebra, the Drinfeld realization of its Yangian is defined in terms of generators and relations. Consequently, the size of the algebra is a priori unclear. Fortunately, for Yangians of a semi-simple Lie algebra, the PBW theorem is well known. It has been conjectured that the PBW theorem holds for a much larger class of Yangians including the affine Yangians. On the other hand, for the (undeformed) current algebra of an affine Lie algebra Enriquez [E03] provided a presentation of the current algebra of the positive Borel subalgebra in terms of Drinfeld-type loop generators and commutation relation. This can be considered as a PBW theorem for the undeformed current algebra of the positive Borel algebra. In the proof of loc. cit., a shuffle algebra description and a duality of the current algebra play the key roles. In the framework of cohomological Hall algebras following the same idea as Schiffmann and Vasserot [SV12], in [YZ14] the authors of the present paper gave a conjectural shuffle algebra description of the Yangian of any symmetric Kac– DOI: 10.1007/S00031-020-09572-6 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. Received August 21, 2018. Accepted September 12, 2019. Corresponding Author: Gufang Zhao, e-mail: [email protected] ∗
YAPING YANG, GUFANG ZHAO
Moody Lie algebra, and constructed a surjective map from the Yangian to the shuffle algebra. In the present paper, we combine this result and the earlier results of Enriquez [E03] to prove the PBW theorem for the Yangian of any untwisted symmetric affine Lie algebra. b n ), this theorem has been proved by N. Guay in [G07]. During the For Y~ (sl preparation of the present paper, Guay, Regelskis, and Wendlandt communicated to the authors an independent proof of the PBW theorem for Y~ (gKM ) using the vertex operator representations of the affine Yangians [GRW18]. This paper is organized as follows. We start by recollecting relevant results from [YZ16]. The statements of the
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