Correspondences Between Cluster Structures.

Abstract

This thesis introduces quasi-homomorphisms of cluster algebras, a class of maps relating cluster algebras of the same type, but with different coefficients. The definition is given in terms of seed orbits, the smallest equivalence classes of seeds on which the mutation rules for non-normalized seeds are unambiguous. After proving basic structural results, we provide examples of quasi-homomorphisms involving familiar cluster algebras. We construct a quasi-isomorphism between cluster stuctures in Grassmannians and cluster structures in Fock-Goncharov spaces of configurations of affine flags. We explore the related notion of a quasi-automorphism, and compare the resulting group with other groups of symmetries of cluster structures. For cluster algebras from surfaces, we determine the subgroup of quasi-automorphisms inside the tagged mapping class group of the surface.