Regular pullback of generalized cluster structures

TL;DR

This paper investigates how to lift regular cluster structures from quasi-affine varieties to affine spaces and defines conditions for a regular pullback, exploring its combinatorics, Poisson brackets, and algebraic properties.

Contribution

It introduces the concept of almost-cluster structures, providing sufficient conditions for their existence and analyzing their combinatorial and algebraic features.

Findings

Established conditions for the existence of almost-cluster structures

Analyzed the combinatorics of the lifted structures

Studied compatible Poisson brackets and upper cluster algebras

Abstract

We consider the problem of lifting a regular cluster structure on a quasi-affine variety to the ambient affine space and a similar problem of defining a regular pullback of a regular cluster structure under a dominant rational map between affine spaces. We provide sufficient conditions for the existence of the corresponding object, called an almost-cluster structure, study its combinatorics, compatible Poisson bracket and the corresponding upper cluster algebra.