Quasi-homomorphisms of cluster algebras

Abstract

Abstract. We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a map between 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. We present examples of quasi-homomorphisms involving familiar cluster algebras, such as cluster structures on Grassmannians, and those associated with marked surfaces with boundary. 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.

Drinfeld double of GLn and generalized cluster structures

“…Following , we call a nerve an arbitrary subtree of on vertices such that all its edges have different labels. A star of a vertex in is an example of a nerve.…”

“…Regularity of the function ϕ 11 that replaces ϕ 11 follows from Proposition 3.5. Let us prove that this function is not divisible by ϕ 11 . Indeed, assume the contrary, and define an n × n matrix Σ 11 via…”

Cluster structures in Schubert varieties in the Grassmannian

“…However, the cluster structure given by Σ v,w is quasi-isomorphic to the cluster structure conjectured in [23,Conjecture 3.4]. This means roughly that seeds in one cluster algebra can be obtained from seeds in the other by rescaling variables by Laurent monomials in frozen variables, in a way that is compatible with mutation (see [10]). Details will appear in work in preparation by Fraser and the second author.…”

Cluster automorphisms and quasi-automorphisms

“…In this section, we recall the notions of cluster automorphisms [2,7] and quasiautomorphisms [9], and we also introduce the notion of weak automorphisms. We study the relation between these notions in section 4.…”

“…Fraser introduced the less restrictive notions of quasi-homomorphisms and quasiautomorphisms in [9]. The definition of a quasi-automorphism does not strictly require that a cluster is mapped to a cluster, but allows for the flexibility that the image of a cluster is a cluster up to rescaling each cluster variable by some coefficients.…”