Publications (2)0 Total impact
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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.
10/2012;
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ABSTRACT: We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of [KR], we prove that any maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via mutations, then their endomorphism algebras have the same representation type. Comment: 14pages, fix many typos, add two references
04/2010;
