Drinfeld double of GLn and generalized cluster structures

Abstract

We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson–Lie group GLn and derive from it a generalized cluster structure on GLn compatible with the push‐forward of the Poisson bracket on the dual Poisson–Lie group.

On the Generalized Cluster Algebras of Geometric Type

“…First, let us recall the definition of the generalized cluster algebras of geometric type (see [4,7]).…”

“…Gekhtman, Shapiro and Vainshtein [7] proved that generalized upper cluster algebras over certain rings retain all properties of ordinary upper cluster algebras, and under certain coprimality conditions coincide with the intersection of rings of Laurent polynomials in a finite collection of clusters.…”

Some elementary properties of Laurent phenomenon algebras

“…Muller showed that locally acyclic cluster algebras coincide with their upper cluster algebras in [11]. Gekhtman, Shapiro and Vainshte in [8] proved (a) for generalized cluster algebras, then Bai, Chen, Ding and Xu demonstrated (c) and the sufficiency of (b) in [1]. Besides, Bai discovered that acyclic generalized cluster algebras coincide with their generalized upper cluster algebras.…”

“…In the theory of cluster algebras, the following are interesting conjectures on seeds of cluster algebras: in a cluster algebra of rank n, (1) each seed is uniquely defined by its cluster; (2) any two seeds with n − 1 common cluster variables are connected with each other by one step of mutation. One can refer [3,8] for detailed proof.…”

“…One should note that our method also works for cluster algebras and generalized cluster algebras. We do not talk much about generalized cluster algebras in this paper, and one can refer [1,3,4,8,12] for details.…”