A $\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver

Details A $\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver A $\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver

2008-11-22

arxiv.org/abs/0811.3676v2

Mathematics

Representation Theory

21 pages

Scientific paper

The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types $\widetilde{A}_{n,n},\widetilde{D}$ and $\widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.

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