Abstract
Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\mathcal {C}$. Especially, for a 2-Calabi-Yau extriangulated category $\mathcal {C}$ with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from $\mathcal {C}$ with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in $\mathcal {C}$ and certain pairs of their rooted cluster subalgebras which we call complete pairs. Finally, we explain this correspondence by an example relating to a Grassmannian cluster algebra.
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