T -structures and torsion pairs in a 2-Calabi-Yau triangulated category

Abstract

For a d-Calabi–Yau triangulated category 𝒷 with a d-cluster tilting subcategory 𝒯, the decomposition of 𝒷 is determined by the decomposition of 𝒯 satisfying ‘vanishing condition’ of negative extension groups, namely, 𝒷=⊕i∈I𝒷i, where 𝒷i, i∈I, are triangulated subcategories, if and only if 𝒯=⊕i∈I𝒯i, where 𝒯i, i∈I, are subcategories with Hom𝒷(𝒯i[t], 𝒯j)=0, ∀ 0⩽t⩽d−2 and i≠j. This induces that for any two cluster tilting objects T and T′ in a 2-Calabi–Yau triangulated category 𝒷, the Gabriel quiver of the endomorphism algebra End𝒷T is connected if and only if so is that of End𝒷T′. As an application, we prove that the 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 torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs. We also discuss the relationship between mutations of torsion pairs and mutations of cluster tilting objects.