arXiv:2105.08211v1 [math.RT] 18 May 2021 Symmetric mutations sub-algebra in the context of sub-seeds Ibrahim Saleh Email: salehi@uww.edu Contents 1 Introduction 1 2 Rooted Cluster Algebras 3 3 Symmetric Sub-mutation algebras 6 4 References 25 Abstract For a rooted cluster algebra A(Q) over a valued quiver, a symmetric cluster variable is any cluster variable belong to a cluster associated to a quiver σ(Q) for some permutation σ. The subalgebra generated by all symmetric cluster variables is called the symmetric mutation subalgebra and is denoted by B(Q). In this paper we identify the class of cluster algebras that satisfy B(Q)= 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 mutations classes based on their weights. Some properties of symmetric mutation subalgebras are given. MSC (2010): Primary 13F60, Secondary 05E15. Keywords: Cluster Algebras, Subseeds, Mutations subalgebras. 1 Introduction Cluster algebras were introduced by S. Fomin and A. Zelevinsky in [5, 6, 2, 7, 15]. A cluster algebra is a commutative ring with a distinguished set of generators called cluster variables which appear in overlapping sets called clusters. Each cluster is paired with a (valued) quiver to form what is so called a seed. A new seed can be obtained from an existing seed using particular type of relations called mutations. The set of all quivers that can be produced from a quiver Q by applying mutations is called mutation class of Q. The mutation class plays a central role in understanding the structure of its associated cluster algebra. In [6], S. Fomin and A. Zelevinsky proved that a cluster algebra is finitely generated if and only if each quiver in its mutation class has weight