A quantum cluster algebra structure on the semi-derived Hall algebra

TL;DR

This paper establishes a quantum cluster algebra structure on the semi-derived Hall algebra of a quiver's projective modules, extending known isomorphisms and actions from quantum loop algebras.

Contribution

It introduces a quantum cluster algebra structure on the semi-derived Hall algebra and constructs a braid group action lifting previous quantum Grothendieck ring actions.

Findings

Quantum cluster algebra structure on semi-derived Hall algebra

Lifting of braid group action to semi-derived Hall algebra

Extension of Hernandez-Leclerc isomorphism

Abstract

Using Hernandez-Leclerc's isomorphism between the derived Hall algebra of a representation-finite quiver $Q$ and the quantum Grothendieck ring of the quantum loop algebra of the Dynkin type of $Q$, we lift the (quantum) cluster algebra structure of the quantum Grothendieck ring to the semi-derived Hall algebra, introduced by Gorsky, of the category of bounded complexes of projective modules over the path algebra of $Q$. We also construct a braid group action on the semi-derived Hall algebra, lifting Kashiwara-Kim-Oh-Park's braid group action on the quantum Grothendieck ring.