关于一类广义丛代数的探讨  

Discussion on a class of generalized cluster algebras

作　　者：李菁菁 陈家辉 LI Jing-jing;CHEN Jia-hui(School of Mathematics and Information Science,Guangzhou University,Guangzhou 510006,China)

机构地区：[1]广州大学数学与信息科学学院,广东广州510006

出　　处：《广州大学学报（自然科学版）》2024年第4期67-76,共10页Journal of Guangzhou University:Natural Science Edition

摘　　要：丛代数是代数学中的一个重要研究对象。它是一类以丛变量为生成元,并且满足某种突变规则的有理函数域中的一个交换子代数。关于秩为2的丛代数A(b,c)已经有了一些研究成果。特别地,当bc≤4时,A(b,c)是有限型丛代数和仿射型丛代数两种类型。现如今,丛代数在泊松几何、表示论、量子群等研究领域中起重要作用。文章研究一类特殊广义丛代数的代数结构,主要通过定义s_(n)(n≥1)来建立此类广义丛代数的乘法公式,并由此证明所有的丛单项式x_(m)^(p)x_(m+1)^(q)(m∈A)和s_(n)(n≥1)组成的集合是此类广义丛代数的一组整基。Cluster algebra is an important research object in algebra.Cluster algebra is a kind of com-mutative subalgebra of a rational function field which satisfies some mutation rule by taking cluster variables as generators.There are some research results on the cluster algebraA(b,c)of rank 2.In particular,when bc≤4,A(b,c)is of two types of algebras with finite type and affine type.Nowa-days,cluster algebra plays an important role in Poisson geometry,representation theory,quantum group and other research fields.This paper studies the algebraic structure of a special class of general-ized cluster algebras,mainly by defining s_(n)(n≥1)to establish the multiplication formula of such gen-eralized cluster algebras,and thus proves that the set of all cluster monomials x_(m)^(p)x_(m+1)^(q)(m∈A)and s_(n)(n≥1)is an integer base of such generalized cluster algebras.

关 键 词：丛代数 丛变量 交换环 广义丛代数 

分 类 号：O187.2[理学—数学]