无环量子丛代数：通过丛范畴相关的Hall代数实现
Acyclic quantum cluster algebras via Hall algebras associated to cluster categories
- SiChuan University（四川大学）
- Nanjing Normal University（南京师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过构造与丛范畴相关的Hall代数，实现了所有无环量子丛代数的Hall代数实现，并证明了量子丛代数同构于该代数的例外子代数，且Dynkin型时自然同构，同时给出了基和突变不变性等应用。
AI中文摘要：
设 $Q$ 为有限无环带值箭图。我们建立了与 $Q$ 相关联的量子丛代数中的另一种丛乘法公式，该公式展现出类似于丛范畴的2-Calabi-Yau性质的对称性。受此公式启发，我们通过 $Q$ 相关的导出Hall子代数的某个商引入了Hall代数 $\mathcal {D}\mathcal {H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$，作为遗传代数丛范畴的代数对应物。作为主要结果，我们证明了与 $Q$ 相关联的量子丛代数同构于 $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ 的例外子代数，即由例外对象生成的子代数。进一步地，当 $Q$ 为Dynkin型时，量子丛代数自然同构于 $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ 本身。这为所有无环量子丛代数建立了Hall代数实现。作为应用，我们引入了由投射对象生成的 $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ 的Hall子代数，给出了下界量子丛代数的Hall代数类比。然后，我们通过Hall代数方法证明了量子投射丛变量的标准单项式构成量子丛代数的一组基。此外，我们证明了丛范畴的Auslander-Reiten平移诱导 $\mathcal {D}\mathcal {H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ 的代数自同构。最后，我们将BGP反射应用于 $\mathcal {D}\mathcal {H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$，并证明了Hall代数 $\mathcal {D}\mathcal {H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ 在 $Q$ 的汇点处突变下同构不变。
英文摘要：
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 $\mathcal {D}\mathcal {H}_Λ^{cl}(\widetilde{\mathcal{A}})$ 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 $\mathcal{D}\mathcal{H}_Λ^{cl}(\widetilde{\mathcal{A}})$, namely, the subalgebra generated by exceptional objects. Furthermore, when $Q$ is of Dynkin type, the quantum cluster algebra is naturally isomorphic to $\mathcal{D}\mathcal{H}_Λ^{cl}(\widetilde{\mathcal{A}})$ itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of $\mathcal{D}\mathcal{H}_Λ^{cl}(\widetilde{\mathcal{A}})$ 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 $\mathcal {D}\mathcal {H}_Λ^{cl}(\widetilde{\mathcal{A}})$. Finally, we apply the BGP-reflections to $\mathcal {D}\mathcal {H}_Λ^{cl}(\widetilde{\mathcal{A}})$, and prove that the Hall algebra $\mathcal {D}\mathcal {H}_Λ^{cl}(\widetilde{\mathcal{A}})$ is invariant up to isomorphism under the mutations at sink vertices of $Q$.
