Locally acyclic cluster algebras

无环箭图的稳定性条件与代数心

Stability Conditions and Algebraic Hearts for Acyclic Quivers

本文研究有限连通无环箭图导出范畴的稳定性条件，证明其心可由代数心经相位旋转得到，从而建立稳定性条件空间的连通性，并证明每个稳定性条件具有完全的例外集合。

Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid

We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O $ and homotopy equivalence classes of negatively graded Lie $\infty $-algebroids over their resolutions (=acyclic Lie $\infty$-algebroids). This extends to a purely algebraic setting the construction of the universal $Q$-manifold of a locally real analytic singular foliation of Lavau-C.L.-Strobl. In particular, it makes sense for the universal Lie $\infty$-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algeb…

On the quiver Grassmannian in the acyclic case

Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.

On the cohomology of quiver Grassmannians for acyclic quivers

For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_q^-(\mathfrak{g})$, where $U_q^-(\mathfrak{g})$ is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result…

A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver

For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The family is parameterized by partitions of the quiver into Dynkin subquivers. For each such partition, we write the domain of our isomorphism as a tensor product of subalgebras in two ways. In the first, each tensor factor is isomorphic to the CoHA of the quiver with a single vertex and no arrows. In the second, the tensor factors are each isomorphic to the CoHAs of the corresponding Dynkin subquivers. When the quiver is already an orientation of a simply-laced Dynkin diagram, o…

簇 DT 变换的熵与无环箭图的有限–驯–野三分法

Entropy of cluster DT transformations and the finite-tame-wild trichotomy of acyclic quivers

本文以簇 DT 变换 τ 的符号稳定性和簇拉伸因子刻画无环箭图的有限–驯–野三分法，并计算相关变异环熵，证明其代数与范畴熵由 Coxeter 矩阵谱半径对数统一给出。

Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. I. Incidence Structure

A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.

Frieze varieties : A characterization of the finite-tame-wild trichotomy for acyclic quivers

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, re…

抽象可定向关联结构与有限有界无环范畴的算法。II. 数据结构与基本操作

Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. II. Data Structure and Fundamental Operations

本文构建了有限有界无环范畴的数据结构以编码抽象可定向关联结构，将其表示为带权有向无环多重图，并从几何、范畴和编程角度研究了其基本操作。

Superpotential algebras and manifolds

In this paper we study a special class of Calabi-Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of `superpotential descriptions' in motivic Donaldson-Thomas theory, we investigate the question of whether these algebras admit superpotential presentations. We establish that the fundamental group algebras of a wide class of acyclic manifolds, including all hyperbolic manifolds, do not admit such descriptions, disproving Ginzburg's conjecture regarding them. We also describe a class of manifolds that…