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