Some elementary properties of Laurent phenomenon algebras

Abstract

<abstract><p>Let $ \Sigma $ be a Laurent phenomenon (LP) seed of rank $ n $, $ \mathcal{A}(\Sigma) $, $ \mathcal{U}(\Sigma) $, and $ \mathcal{L}(\Sigma) $ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $ \mathcal{A}(\Sigma) $ is uniquely defined by its cluster and any two seeds of $ \mathcal{A}(\Sigma) $ with $ n-1 $ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $ \mathcal{U}(\Sigma) $ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $ \Sigma $. Besides, we obtain the standard monomial bases of $ \mathcal{L}(\Sigma) $. We also prove that $ \mathcal{U}(\Sigma) $ coincides with $ \mathcal{L}(\Sigma) $ under certain conditions.</p></abstract>