Symmetric mutations algebras in the context of sub-cluster algebras

TL;DR

This paper studies symmetric mutation subalgebras within cluster algebras, identifying when they coincide with the entire algebra and classifying quivers based on their mutation classes and weights.

Contribution

It characterizes cluster algebras where symmetric mutation subalgebras equal the whole algebra and classifies quivers by their mutation classes and weights.

Findings

Identifies conditions for symmetric mutation subalgebras to equal the entire cluster algebra.

Provides a classification of quivers based on mutation classes and weights.

Shows that almost all finite mutation type quivers satisfy the main condition.

Abstract

For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra of $\mathcal{A}(Q)$ generated by all symmetric cluster variables is called the \emph{symmetric mutation subalgebra} and is denoted by $\mathcal{B}(Q)$. In this paper we identify the class of cluster algebras that satisfy $\mathcal{B}(Q)=\mathcal{A}(Q)$, which contains almost every quiver of finite mutation type. In the process of proving the main theorem, we provide a classification of quivers mutation classes based on their weights. Some properties of symmetric mutation subalgebras are given.