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Relation between generalized and ordinary cluster algebras

Ryota Akagi, Tomoki Nakanishi

TL;DR

This work extends the partial realization of generalized cluster algebras (GCAs) to GCAs with $y$-variables in arbitrary semifields by developing composite cluster patterns that lift GCAs to ordinary cluster algebras. The authors construct enlargements of exchange matrices and composite mutations, then realize GCAs as subquotients of standard cluster algebras via a carefully controlled quotient, ensuring a prime relation ideal and consistent factorization of exchange polynomials. They establish precise correspondences between $C$- and $G$-matrices and $F$-polynomials for GCAs and their composite counterparts through separation formulas and recursive mutations, providing concrete realizations and field isomorphisms. The results unify generalized and ordinary cluster theory in a broad semifield context and yield explicit mechanisms to transfer structural data between GCAs and composite/ordinary cluster patterns. Overall, the paper significantly clarifies the internal structure of GCAs and broadens their applicability by linking them to classical cluster algebras through a robust, algebraic, and combinatorial framework.

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.

Relation between generalized and ordinary cluster algebras

TL;DR

This work extends the partial realization of generalized cluster algebras (GCAs) to GCAs with -variables in arbitrary semifields by developing composite cluster patterns that lift GCAs to ordinary cluster algebras. The authors construct enlargements of exchange matrices and composite mutations, then realize GCAs as subquotients of standard cluster algebras via a carefully controlled quotient, ensuring a prime relation ideal and consistent factorization of exchange polynomials. They establish precise correspondences between - and -matrices and -polynomials for GCAs and their composite counterparts through separation formulas and recursive mutations, providing concrete realizations and field isomorphisms. The results unify generalized and ordinary cluster theory in a broad semifield context and yield explicit mechanisms to transfer structural data between GCAs and composite/ordinary cluster patterns. Overall, the paper significantly clarifies the internal structure of GCAs and broadens their applicability by linking them to classical cluster algebras through a robust, algebraic, and combinatorial framework.

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 -variables in an arbitrary semifield. We also present the relations between the -matrices, the -matrices, and the -polynomials of a generalized cluster pattern and those of the corresponding composite cluster pattern.
Paper Structure (18 sections, 16 theorems, 64 equations, 2 tables)

This paper contains 18 sections, 16 theorems, 64 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Generalized cluster algebras with $y$-variables
  3. Semifields
  4. Generalized seeds and GCAs
  5. Composite cluster patterns
  6. Enlargements of exchange matrices
  7. Composite cluster patterns
  8. Realization of generalized $Y$-pattern
  9. Extensions of semifields
  10. Realization of generalized $Y$-patterns
  11. Realization of generalized cluster algebras
  12. Realization of generalized cluster patterns
  13. Realization of GCAs
  14. Relations for $C$-matrices, $G$-matrices, and $F$-polynomials
  15. Separation formulas
  16. ...and 3 more sections

Key Result

Lemma 3.4

For a generalized cluster pattern with mutation degree ${\bf r}$, let $\{B_{t}\}_{t \in \mathbb{T}_n}$ be the collection of its exchange matrices. Let $\mathcal{B}_{t}$ be the enlargement of $B_t$ with ${\bf r}$. Then, for any pair of $k$-adjacent vertices $t,t' \in \mathbb{T}_n$, we have

Theorems & Definitions (41)

  • Definition 2.1: Semifield
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Generalized cluster algebra
  • Definition 3.1: Enlargement
  • Example 3.2
  • Definition 3.3: Composite mutation
  • Lemma 3.4: RW25
  • proof
  • Lemma 3.5
  • ...and 31 more