  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.
