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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 thedecomposition 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 $\mathcal{T}_i, i\in I$ aresubcategories 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 quiversof endomorphism algebras of any two cluster tilting objects in a $2-$Calabi-Yautriangulated category are connected or not at the same time. As an application,we prove that indecomposable $2-$Calabi-Yau triangulated categories withcluster tilting objects have no non-trivial t-structures and no non-trivialco-t-structures. This allows us to give a classification of cotorsion pairs inthis triangulated category. Moreover the hearts of cotorsion pairs in the senseof Nakaoka are equivalent to the module categories over the endomorphismalgebras of the cores of the cotorsion pairs.

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