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arXiv · 1707.04661

Extending Upper Cluster Algebras

Jiarui Fei·Jerzy Weyman
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Abstract

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the restriction of $(Δ,W)$ to some subquiver is a cluster model, we give a sufficient condition for $(Δ,W)$ itself being a cluster model. As an application, we show that the semi-invariant ring of any complete $m$-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.

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Jiarui Fei, Jerzy Weyman. 2017-07-14. Extending Upper Cluster Algebras. https://arxiv.org/abs/1707.04661

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