Special folding of quivers and cluster algebras

We give a precise definition of folded quivers and folded cluster algebras. We define a special folding of a quiver as one which cannot be associated with a skew-symmetrizable exchange matrix. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite mutation type quivers X₆ and X₇. We also construct a folded cluster algebra associated to punctured surfaces which allow for self-folded triangles. We give a simple construction of a folded cluster algebra for which the cluster complex is a generalized permutohedron.