The main theorem is substantially the same, but we added Section 4 on an important variant of the construction which admits a Hopf algebra structure, and Section 5 on non-generic parameters. Several applications are discussed in Subsections 1.3 and 1.5
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo-Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized K-theoretic Hall algebra associated to the quiver by Schiffmann-Vasserot. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in Neguţ-Sala-Schiffmann to give a generators-and-relations description of the Hall algebra of any curve over a finite field (which is a shuffle algebra due to Kapranov-Schiffmann-Vasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized K-theoretic Hall algebra, and the positive half of the corresponding quantum loop group are all isomorphic; we also obtain the non-degeneracy of the Hopf pairing on the latter quantum loop group.
Folding shuffle algebras and twisted $q$-characters
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Simion Stoilow Institute of Mathematics (IMAR)（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）
