Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category

TL;DR

This paper investigates the structure of 2-Calabi-Yau triangulated categories, establishing conditions for their decomposition, classifying cotorsion pairs, and analyzing the properties of t-structures and hearts in this context.

Contribution

It provides a classification of cotorsion pairs in 2-Calabi-Yau triangulated categories and shows the absence of non-trivial t-structures and co-t-structures in indecomposable cases.

Findings

Decomposition of categories is linked to special decompositions of cluster tilting subcategories.

Gabriel quivers of endomorphism algebras are either all connected or all disconnected for different cluster tilting objects.

Indecomposable 2-Calabi-Yau categories with cluster tilting objects have no non-trivial t-structures or co-t-structures.

Abstract

For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of $\mathcal{T}$, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that the Gabriel quivers of endomorphism algebras of any two cluster tilting objects in a $2-$Calabi-Yau triangulated category are connected or not at the same time. As an application, we prove that indecomposable $2-$Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of cotorsion pairs in this triangulated category. Moreover the hearts of cotorsion pairs in the sense of Nakaoka are equivalent to the module categories over the endomorphism algebras of the cores of the cotorsion pairs.