Polynomial recognition of cluster algebras of finite type

JOURNAL OF ALGEBRA

卷:457

Article

Dias, Elisangela Silva¹  Castonguay, Diane¹ 

[1] Univ Fed Goias, Inst Informat, Alameda Palmeiras,Campus Samambaia, BR-74001970 Goiania, Go, Brazil

关键词: Cluster algebra;    Finite type;    Algorithm;   

DOI  :  10.1016/j.jalgebra.2016.03.027

来源: Elsevier

【 摘 要 】

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky [1] presented an easier way to verify if a given algebra is of finite type, by testing if all chordless cycles of the graph related to the algebra are cyclically oriented and if there exists a positive quasi-Cartan companion of the skew-symmetrizable matrix related to the algebra. We develop an algorithm that verifies these conditions and decides whether or not a cluster algebra is of finite type in polynomial time. The second part of the algorithm is used to prove that the more general problem to decide if a matrix has a positive quasi-Cartan companion is in NP class. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free