For a fixed seed $(X, Q)$, a \emph{ rooted mutation loop} is a sequence of mutations that preserves $(X, Q)$. The group generated by all rooted mutation loops is called \emph{ rooted mutation group} and will be denoted by $\mathcal{ M}(Q)$. The \emph{ global mutation group} of $(X, Q)$, denoted $\mathcal{ M}$, is the group of all mutation sequences subject to the relations on the cluster structure of $(X, Q)$. In this article, we show that two finite type cluster algebras $\mathcal{ A}(Q)$ and $\mathcal{ A}(Q')$ are isomorphic if and only if their rooted mutation groups are isomorphic and the sets $\mathcal{ M}/\mathcal{ M}(Q)$ and $\mathcal{ M'}/\mathcal{ M}(Q')$ are in one to one correspondence. The second main result shows that the group $\mathcal{ M}(Q)$ and the set $\mathcal{ M}/\mathcal{ M}(Q)$ determine the finiteness of the cluster algebra $\mathcal{ A}(Q)$ and vice versa.
Categories: math.RT