Shuffle algebras for quivers and wheel conditions

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 Negu\c{t}-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.