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arXiv · 2410.18604

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

Alessandro Contu
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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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Alessandro Contu. 2024-10-24. A quantum cluster algebra structure on the semi-derived Hall algebra. https://arxiv.org/abs/2410.18604

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