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Paper IPM / M / 17372  


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 socalled 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 coefficientfree cluster algebra of types Anor Anis equal to the corresponding lower bound cluster algebra generated by projective cluster variables.
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