Spherical objects and stability conditions on 2-Calabi--Yau quiver categories

有限维Mesh代数的对称性、周期和Calabi-Yau维数

The symmetry, period and Calabi-Yau dimension of finite dimensional mesh algebras

在有限维网格代数类，也称为m-折叠网格代数，我们确定那些是对称的，那些稳定的模范畴是弱卡-丘。我们还给出了，在组合方面，明确的公式期间的任何这样的代数，并为卡拉比-丘弗罗贝纽斯和稳定的卡拉比-丘维，当他们被定义。

Erratum to "Deformed Calabi-Yau completions"

We correct an error and some inaccuracies that occurred in "Deformed Calabi-Yau completions''. The most important point is that, as pointed out by W. K. Yeung, to show that the deformed Calabi-Yau completion has the Calabi-Yau property, one needs to assume that the deformation parameter comes from negative cyclic homology. Notice that this does hold in the case of Ginzburg dg algebras.

具有簇倾斜子范畴的Calabi-Yau范畴的结构

On the structure of Calabi-Yau categories with a cluster tilting subcategory

证明了对于$d \geq 2$，一个赋$d$ -簇倾斜子范畴的代数$d$ -Calabi-Yau三角范畴是一个DG范畴的稳定范畴，该DG范畴完全是$(d+1)$ -Calabi-Yau，并且具有一个非退化的$t$ -结构，其中心具有足够的投射.

2-Calabi-Yau三角范畴中倾斜簇对象的提升

Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories

我们证明了在2-Calabi-Yau三角范畴中的一个簇倾斜对象的自同态代数上的倾斜模提升到这个2-Calabi-Yau三角范畴中的一个簇倾斜对象。这推广了D.史密斯的聚类分类。

相对卡丘结构

Relative Calabi-Yau structures

引入定义在微分分次范畴函子上的相对非交换Calabi-Yau结构。例子出现在各种背景下，如拓扑，代数几何，和表示论。我们的主要结果是一个Calabi-Yau余跨距的合成律，推广了定向流形的余边的经典合成。作为应用，我们构造了框架穿孔黎曼曲面的拓扑范畴上的Calabi-Yau结构。

有限型自内射代数的稳定Calabi-Yau维数

Stable Calabi--Yau dimension of self-injective algebras of finite type

利用双模合偶和所谓的稳定内自同构给出了稳定Calabi-Yau维数的一个等价定义.利用它，我们完成了由K. Erdmann，A. Skowroplski，J. Bia \lkowski和A.杜加斯

Gorenstein代数和稳定Calabi-Yau代数

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

证明了在2-Calabi-Yau三角范畴中，每个簇倾斜子范畴都是Gorenstein的，它的每个簇生成的内射维数至多为1。我们证明了它的Cohen-Macaulay模的稳定范畴是3-Calabi-Yau。我们特别推出，集群倾斜代数是Gorenstein的维数最多为1，遗传，如果他们是有限的整体维数。我们的结果也适用于稳定的（！）Geiss-Leclerc-Schroer极大刚性模的自同态环此外，我们还证明了非稳定自同态环上相对3-Calabi-Yau对偶的一个一般结果。这加强和推广了Geiss-Leclerc-Schroer对简单模所得到的Ext群对称性。最后，我们将2-Calabi-Yau范畴的相对Calabi-Yau对偶的结果推广到d-Calabi-Yau范畴。我们展示了如何产生许多d-簇倾斜代数的例子。

Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

We show that a Calabi-Yau structure of dimension $d$ on a smooth dg category $C$ induces a symplectic form of degree $2-d$ on the moduli space of objects $M_{C}$. We show moreover that a relative Calabi-Yau structure on a dg functor $C \to D$ compatible with the absolute Calabi-Yau structure on $C$ induces a Lagrangian structure on the corresponding map of moduli $M_{D} \to M_{C}$.

Pre-Calabi-Yau structures and moduli of representations

We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau structure on a non-positively graded dg algebra $A$ induces a $(2-n)$-shifted symplectic structure on its derived moduli stack of representations; while any $n$-pre-Calabi-Yau structure on $A$ induces a $(2-n)$-shifted Poisson structure on this derived moduli stack.