Quasi-homomorphisms of cluster algebras

TL;DR

This paper introduces the concept of quasi-homomorphisms in cluster algebras, providing a flexible framework for maps between cluster algebras with different coefficients and exploring their symmetries.

Contribution

It defines quasi-homomorphisms via seed orbits, presents examples involving Grassmannians and surfaces, and analyzes the quasi-automorphism group within the context of cluster algebra symmetries.

Findings

Defined quasi-homomorphisms using seed orbits

Identified quasi-automorphisms in cluster algebras from surfaces

Determined subgroup of quasi-automorphisms inside the tagged mapping class group

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.