Symmetric mutations algebras in the context of sub-cluster algebras

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 $σ(Q)$, for some permutation $σ$. 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.

Theorems & Definitions (30)

• Definition 2.2: Valued quiver mutations
• Definition 2.4: Seed mutation
• Definition 2.6
• Definition 2.7
• Example 2.10
• Theorem 3.5
• Lemma 3.6
• proof
• Definition 3.7
• Proposition 3.9