Discussion on a class of generalized cluster algebras

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Journal of Guangzhou University(Natural Science Edition) ›› 2024, Vol. 23 ›› Issue (4) : 67-66.

Discussion on a class of generalized cluster algebras

  • LI Jing jing, CHEN Jia hui
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Abstract

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 algebra A( 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. Nowadays, 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 generalized cluster algebras, mainly by defining sn ( n≥1) to establish the multiplication formula of such generalized cluster algebras, and thus proves that the set of all cluster monomials xpm xqm+1 ( m∈Z) and sn( n≥1) is an integer base of such generalized cluster algebras.

Key words

cluster algebra / cluster variable / commutative ring / generalized cluster algebra

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Discussion on a class of generalized cluster algebras. Journal of Guangzhou University(Natural Science Edition). 2024, 23(4): 67-66

References

[16]ShermanP,ZelevinskyA. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types[J]. Moscow Mathematical Journal,2003,4(4):947974.
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