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Shuffle algebras for quivers as quantum groups

Andrei Neguţ, Francesco Sala, Olivier Schiffmann

Abstract

We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], 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 [SV12]). 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 [KSV17]).

Shuffle algebras for quivers as quantum groups

Abstract

We define a quantum loop group associated to an arbitrary quiver and maximal set of deformation parameters, with generators indexed by and some explicit quadratic and cubic relations. We prove that is isomorphic to the (generic, small) shuffle algebra associated to the quiver and hence, by [Neg21a], to the localized K-theoretic Hall algebra of . For the quiver with one vertex and loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus (invoking the results of [SV12]). 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 [KSV17]).

Paper Structure

This paper contains 7 sections, 20 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. The big shuffle algebra and the quadratic quantum loop group
  3. The small shuffle algebra and the quantum loop group
  4. The Drinfeld double of $\mathbf{U}^+$
  5. Special values of the parameters
  6. The spherical Hall algebra of a generic curve
  7. The whole Hall algebra of an arbitrary curve

Key Result

Theorem 1.3

There is an algebra isomorphism $K_{Q,\emph{loc}} \simeq \mathbf{U}^+_Q$.

Theorems & Definitions (48)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.7
  • Remark 2.2
  • Definition 2.4
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.9
  • Remark 2.10
  • ...and 38 more