关于带系数的簇代数和2-Calabi-Yau范畴

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

Within the class of finite dimensional mesh algebras, also called m-fold mesh algebras, we identify those which are symmetric and those whose stable module category is weakly Calabi-Yau. We also give, in combinatorial terms, explicit formulas for the period of any such algebra, and for the Calabi-Yau Frobenius and stable Calabi-Yau dimensions, when they are defined.

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.

A non-simply laced version for cluster structures on 2-Calabi-Yau categories

This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting subcategory with neither loops nor 2-cycles do have the generalized version of cluster structure. This is in particular the case of cluster categories over non-algebraically closed fields.

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

We give an equivalent definition of the stable Calabi--Yau dimension in terms of bimodule syzygies and so-called stably inner automorphisms. Using it, we complete the computation of the stable Calabi--Yau dimensions of the self-injective algebras of finite representation type which was started by K. Erdmann, A. Skowroński, J. Bia\lkowski and A. Dugas.

Relative Calabi-Yau structures

We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi-Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi-Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

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 $M_{D} \to M_{C}$.

The interplay between 2-and-3-Calabi--Yau triangulated categories

This short note surveys the constructions of 3-Calabi--Yau triangulated categories with simple-minded collections due to Ginzburg and Kontsevich--Soibelman and the constructions of 2-Calabi--Yau triangulated categories with cluster-tilting objects due to Buan--Marsh--Reineke--Reiten--Todorov and Amiot, and includes a discussion on the normal form of 2-Calabi--Yau triangulated categories with cluster-tilting objects.

Decorated Marked Surfaces: Calabi-Yau categories and related topics

This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration $Δ$ on a marked surfaces $\mathbf{S}$, to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.

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.