Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
Abstract
Let Q be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with Q, which exhibits certain symmetries similar to 2-Calabi--Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra 𝒟ℋΛcl(𝒜̃) via a certain quotient of a derived Hall subalgebra associated with Q, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with Q is isomorphic to the exceptional subalgebra of 𝒟ℋΛcl(𝒜̃), namely, the subalgebra generated by exceptional objects. Furthermore, when Q is of Dynkin type, the quantum cluster algebra is naturally isomorphic to 𝒟ℋΛcl(𝒜̃) itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of 𝒟ℋΛcl(𝒜̃) generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander--Reiten translation of cluster categories induces an algebra automorphism of 𝒟ℋΛcl(𝒜̃). Finally, we apply the BGP-reflections to 𝒟ℋΛcl(𝒜̃), and prove that the Hall algebra 𝒟ℋΛcl(𝒜̃) is invariant up to isomorphism under the mutations at sink vertices of Q.
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BibTeX
@misc{fu2026acyclic,
title = {{Acyclic quantum cluster algebras via Hall algebras associated to cluster categories}},
author = {Changjian Fu and Haicheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14388},
eprint = {2609.14388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14388},
}
RIS
TY - UNPB
TI - Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
AU - Changjian Fu
AU - Haicheng Zhang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14388
UR - https://arxiv.org/abs/2609.14388
AB - Let Q be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with Q, which exhibits certain symmetries similar to 2-Calabi--Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra 𝒟ℋ_Λ^cl(𝒜̃) via a certain quotient of a derived Hall subalgebra associated with Q, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with Q is isomorphic to the exceptional subalgebra of 𝒟ℋ_Λ^cl(𝒜̃), namely, the subalgebra generated by exceptional objects. Furthermore, when Q is of Dynkin type, the quantum cluster algebra is naturally isomorphic to 𝒟ℋ_Λ^cl(𝒜̃) itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of 𝒟ℋ_Λ^cl(𝒜̃) generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander--Reiten translation of cluster categories induces an algebra automorphism of 𝒟ℋ_Λ^cl(𝒜̃). Finally, we apply the BGP-reflections to 𝒟ℋ_Λ^cl(𝒜̃), and prove that the Hall algebra 𝒟ℋ_Λ^cl(𝒜̃) is invariant up to isomorphism under the mutations at sink vertices of Q.
ER -