  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.
