Some elementary properties of Laurent phenomenon algebras

Abstract

Let Σ be Laurent phenomenon (LP) seed of rank n, A(Σ), U (Σ) and L(Σ) be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively.We prove that each seed of A(Σ) is uniquely defined by its cluster, and any two seeds of A(Σ) 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 U (Σ) is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of Σ. Besides, we obtain the standard monomial bases of L(Σ). We also prove that U (Σ) coincides with L(Σ) under certain conditions.

Generalized quantum cluster algebras: the Laurent phenomenon and upper bounds

“…In [2], we proved that generalized cluster algebras coincide with upper bounds under the conditions of acyclicity and coprimality. Du and Li [8] proved that the above results also hold in Laurent phenomenon algebras under some certain conditions.…”