Relation between generalized and ordinary cluster algebras

TL;DR

This paper extends the understanding of generalized cluster algebras by demonstrating their relation to ordinary cluster algebras in broader contexts and exploring their matrix and polynomial relations.

Contribution

It generalizes previous results by showing that all generalized cluster algebras with $y$-variables in any semifield relate to ordinary cluster algebras, and clarifies their matrix and polynomial relations.

Findings

Generalized cluster algebras with $y$-variables in any semifield are isomorphic to quotients of subalgebras of cluster algebras.

Relations between $C$-matrices, $G$-matrices, and $F$-polynomials are established.

The property previously known for geometric type extends to more general cases.

Abstract

Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any generalized cluster algebra with $y$-variables in an arbitrary semifield. We also present the relations between the $C$-matrices, the $G$-matrices, and the $F$-polynomials of a generalized cluster pattern and those of the corresponding composite cluster pattern.