Cluster algebras generated by projective cluster variables

TL;DR

This paper introduces a new class of cluster algebras generated by projective cluster variables, establishing conditions under which they coincide with the original algebra and constructing bases for certain types.

Contribution

It defines lower bound cluster algebras generated by projective variables and proves their equivalence to original algebras under acyclicity, also constructing bases for specific types.

Findings

Lower bound cluster algebra equals the original under acyclicity.

Constructed bases for cluster algebras of types A_n and in_{A_n}.

Established equality of coefficient-free cluster algebras and lower bound algebras for certain types.

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 $\widetilde{A}_n$ is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.