Shuffle algebras for quivers as quantum groups

Abstract

We define a quantum loop group U Q + associated to an arbitrary quiver Q = ( I , E ) and maximal set of deformation parameters, with generators indexed by I × Z and some explicit quadratic and cubic relations. We prove that U Q + is isomorphic to the (generic, small) shuffle algebra associated to the quiver Q and hence, by Neguț (Shuffle algebras for quivers and wheel conditions. arXiv:2102.11269 ), to the localized K-theoretic Hall algebra of Q. For the quiver with one vertex and g loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus g [invoking the results of Schiffmann and Vasserot (Math Ann 353(4):1399–1451, 2012)]. We extend the above results to the case of non-generic parameters satisfying a certain natural metric condition. As an application, we obtain a description by generators and relations of the subalgebra generated by absolutely cuspidal eigenforms of the Hall algebra of an arbitrary smooth projective curve [(invoking the results of Kapranov et al. (Sel Math (NS) 23(1):117–177, 2017)].