Cluster algebras generated by projective cluster variables

DOI10.1016/J.JALGEBRA.2023.02.027zbMATH Open1529.13016arXiv2011.03720MaRDI QIDQ6353202

Publication date: 7 November 2020

Abstract: We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types

A_{n}

or

w i d e t i l d e A_{n}

is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Representations of quivers and partially ordered sets (16G20) Cluster algebras (13F60)

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