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Some properties of generalized cluster algebras of geometric types

Junyuan Huang, Xueqing Chen, Fan Xu, Ming Ding

Abstract

We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric types. We prove that this lower bound algebra coincides with the corresponding generalized cluster algebra under the coprimality condition. As a corollary, we obtain the dual PBW bases of these generalized cluster algebras. Moreover, we show that if the standard monomials of a generalized cluster algebra of geometric type are linearly independent, then the directed graph associated to the initial generalized seed of this generalized cluster algebra does not have 3-cycles.

Some properties of generalized cluster algebras of geometric types

Abstract

We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric types. We prove that this lower bound algebra coincides with the corresponding generalized cluster algebra under the coprimality condition. As a corollary, we obtain the dual PBW bases of these generalized cluster algebras. Moreover, we show that if the standard monomials of a generalized cluster algebra of geometric type are linearly independent, then the directed graph associated to the initial generalized seed of this generalized cluster algebra does not have 3-cycles.
Paper Structure (4 sections, 6 theorems, 41 equations)

This paper contains 4 sections, 6 theorems, 41 equations.

Table of Contents

  1. Background
  2. Preliminaries
  3. The dual PBW bases
  4. A criteria of generalized cluster algebras without 3-cycles

Key Result

Theorem 2.7

BCDX-1 Let $( {\widetilde{\mathbf{x}},\rho,{\widetilde{B}}})$ is an acyclic and coprime generalized seed, then the set of standard monomials in ${x_1},{x'_1}, \ldots ,{x_n},{x'_n}$ is a $\mathbb{Z}\mathbb{P}$-basis of the generalized cluster algebra $\mathcal{A}(\widetilde{\mathbf{x}},\rho,\widetild

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 17 more