Noncommutative marked surfaces II: tagged triangulations, clusters, and their symmetries

TL;DR

This paper develops a noncommutative cluster algebra framework for marked surfaces with orbifold points, introducing new symmetries, tagged clusters, and proving a noncommutative Laurent phenomenon.

Contribution

It extends cluster algebra theory to noncommutative settings on complex surfaces, including orbifolds, with new symmetries and structural results.

Findings

Defined noncommutative cluster structures on surface-related algebras

Constructed new symmetries and tagged clusters for punctured surfaces

Established a noncommutative Laurent Phenomenon

Abstract

The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a given group $G$ into the multiplicative monoid ${\mathcal A}^\times$ and an action of a certain braid-like group $Br_{\mathcal A}$ by automorphisms of each cluster group in a compatible way. For punctured surfaces we construct new symmetries, noncommutative tagged clusters and establish a noncommutative Laurent Phenomenon.