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Generalized cluster algebras are subquotients of cluster algebras

Rolando Ramos, David Whiting

TL;DR

This work resolves the generalized cluster algebra embedding question by constructing a two-pronged approach: homogenize a given GCA via adjoining nth roots of frozen variables and then realize the GCA as a subquotient of a classical CA through an unfolding $\mathcal{B}$ and a folding to a quotient CA. The authors introduce and leverage Hadamard and double-constant block conditions to couple mutations of the original exchange matrix $B$ with group mutations of $\mathcal{B}$, and they define a quotient by an ideal $\mathcal{I}$ that preserves the generalized coefficients as roots of exchange polynomials. The main achievement is a rigorous embedding theorem: there exists a $\mathbb{ZP}$-algebra embedding $\Phi$ from the GCA into a CA, with the GCA realized as a subquotient after scalar restriction, thereby bridging GCAs and classical cluster theory. This construction broadens the scope of CA techniques for GCAs and provides a concrete pathway to transfer properties and methods between the two frameworks.

Abstract

Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified version of this conjecture by providing a construction that realizes a given GCA as subquotient of some CA, as an algebra over the ground ring of the GCA via restriction of scalars.

Generalized cluster algebras are subquotients of cluster algebras

TL;DR

This work resolves the generalized cluster algebra embedding question by constructing a two-pronged approach: homogenize a given GCA via adjoining nth roots of frozen variables and then realize the GCA as a subquotient of a classical CA through an unfolding and a folding to a quotient CA. The authors introduce and leverage Hadamard and double-constant block conditions to couple mutations of the original exchange matrix with group mutations of , and they define a quotient by an ideal that preserves the generalized coefficients as roots of exchange polynomials. The main achievement is a rigorous embedding theorem: there exists a -algebra embedding from the GCA into a CA, with the GCA realized as a subquotient after scalar restriction, thereby bridging GCAs and classical cluster theory. This construction broadens the scope of CA techniques for GCAs and provides a concrete pathway to transfer properties and methods between the two frameworks.

Abstract

Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified version of this conjecture by providing a construction that realizes a given GCA as subquotient of some CA, as an algebra over the ground ring of the GCA via restriction of scalars.
Paper Structure (10 sections, 8 theorems, 82 equations, 3 figures)

This paper contains 10 sections, 8 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result
  4. Adjoining nth roots of frozen variables to a GCA $\mathcal{A}(\mathbf{x},B,\mathcal{P})$
  5. Construction of an unfolding $\mathcal{B}$ of $B$
  6. Construction of $\mathcal{B}$
  7. Group mutation class of $\mathcal{B}$
  8. The Folded Cluster Algebra associated to $\mathcal{B}$
  9. Group Mutation Equivalent Seeds
  10. The Embedding

Key Result

Theorem 1

Any Generalized Cluster Algebra is a Subquotient of a Cluster Algebra.

Figures (3)

  • Figure 1: A given node weighted quiver $Q$ of a modified exchange matrix $\hat{B}$ and the constructed the node weighted quiver $\overline{Q}$ whose exchange polynomials are homogenous without floor function as coefficient.
  • Figure 2: A special case of our construction of $\mathcal{B}$ when it has an associated quiver $\mathcal{Q}$.
  • Figure 3: The exchange matrices from figure (\ref{['fig: matrix construction good figure']}) after a sequence of mutations and the corresponding sequence of group mutations. Here the non-mutatable or slack columns of the exchange matrix $B$ and its associated sub-matrix of $\mathcal{B}$ are red.

Theorems & Definitions (38)

  • Theorem 1
  • Definition 2
  • Definition 3: Generalized seed (of geometric type)
  • Remark 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Definition 8
  • Definition 9: Node Weighted Quiver
  • Definition 10: Folding of a Quiver and Group Mutations
  • ...and 28 more