Geometric construction of cluster algebras and cluster categories

箭图表示与 Hecke 范畴和 Khovanov 弧代数的同构

Quiver presentations and isomorphisms of Hecke categories and Khovanov arc algebras

本文证明了扩展 Khovanov 弧代数与对称群极大抛物子群的反球面 Hecke 范畴基本代数同构，并用箭图与关系给出表示及 Verma 模的完整子模格。

Cluster categories

Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for phenomena occurring in the context of algebraic groups. Cluster algebras also have links to a wide range of other subjects, including the representation theory of finite dimensional algebras, as first discovered by Marsh- Reineke-Zelevinsky. Modifying module categories over hereditary algebras, cluster categories were introduced in work with Buan-Marsh-Reineke-Todorov in order to "categorify" the essential ingredients in the definition of cluster algebras in the acyclic case. They were s…

Mutation of torsion pairs in triangulated categories and its geometric realization

We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.

Geometric model for module categories of Dynkin quivers via hearts of total stability conditions

We derive a geometric model for the module category $\operatorname{mod} \mathbf{k} Q$ of a Dynkin quiver $Q$ via the heart of a total stability condition on the bounded derived category of $\operatorname{mod} \mathbf{k} Q$. As an application, we prove Reineke's conjecture that there is a stability function on $\operatorname{mod} \mathbf{k} Q$ making any indecomposables stable.

Category Algebras and States on Categories

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functionals defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that states on categories as linear functionals defined on category algebras turn out to be generalized of probability measures on sets as discrete categories. Moreover, by establishing a generalizatio…