K-theoretic Hall algebras, quantum groups and super quantum groups

TL;DR

This paper establishes an isomorphism between the K-theoretic Hall algebra of affine preprojective algebras and quantum toroidal groups, and explores connections with super quantum groups and Landau-Ginzburg models.

Contribution

It proves the isomorphism between K-theoretic Hall algebras and quantum groups, and compares super toroidal quantum groups with Hall algebras of quivers with potential.

Findings

K-theoretic Hall algebra is isomorphic to the positive half of a quantum toroidal quantum group.

Deformation of the Hall algebra is torsion free over a polynomial subalgebra.

Comparison between super toroidal quantum groups and Hall algebras of quivers with potential.

Abstract

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the deformation is torsion free over some polynomial subalgebra. Next, we compare super toroidal quantum groups of type A with K-theoretic Hall algebras of quivers with potential, which are defined using the Grothendieck groups of categories of singularities of some Landau-Ginzburg models.