Shuffle algebras for quivers and wheel conditions

TL;DR

This paper proves that the shuffle algebra for doubled quivers is generated by minimal degree elements and is isomorphic to the localized K-theoretic Hall algebra, with implications for quantum groups and Hall algebras of curves.

Contribution

It establishes the generation of the shuffle algebra by minimal degree elements and its isomorphism to the localized K-theoretic Hall algebra, extending to specializations and connections with quantum groups.

Findings

Shuffle algebra generated by minimal degree elements

Isomorphism with localized K-theoretic Hall algebra

Non-degeneracy of the Hopf pairing in quantum loop groups

Abstract

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.