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Motivic cluster multiplication formulas in 2-Calabi-Yau categories

Jie Xiao, Fan Xu, Fang Yang

Abstract

We introduce a notion of motivic cluster characters via virtual Poincaré polynomials, and prove a motivic version of multiplication formulas obtained by Chen-Xiao-Xu for weighted quantum cluster characters associated to a 2-Calabi-Yau triangulated category $\mathcal{C}$ with a cluster tilting object. Furthermore, a refined form of this formula is also given. When $\mathcal{C}$ is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in [International Mathematics Research Notices, rnad172(2023)].

Motivic cluster multiplication formulas in 2-Calabi-Yau categories

Abstract

We introduce a notion of motivic cluster characters via virtual Poincaré polynomials, and prove a motivic version of multiplication formulas obtained by Chen-Xiao-Xu for weighted quantum cluster characters associated to a 2-Calabi-Yau triangulated category with a cluster tilting object. Furthermore, a refined form of this formula is also given. When is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in [International Mathematics Research Notices, rnad172(2023)].
Paper Structure (13 sections, 27 theorems, 151 equations)

This paper contains 13 sections, 27 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. 2-Calabi-Yau categories and motivic multiplication formulas
  3. Exact structure
  4. Virtual Poincaré polynomials
  5. Motivic weighted cluster characters
  6. Motivic weighted multiplication formulas
  7. The refined motivic weighted multiplication formula
  8. Refined motivic multiplication formulas in hereditary case
  9. Cluster category of an acyclic quiver
  10. Specialized quantum cluster characters
  11. The refined multiplication formula in hereditary case
  12. A refined multiplication formula for V of dimension 1
  13. Cluster multiplication formulas with initial cluster characters

Key Result

Lemma 2.1

Fix $(M_0,N_0)\in {\mathrm{Gr}}_e(FM)\times {\mathrm{Gr}}_f(FN)$. (i) For $\epsilon\in \mathrm{Ext}_{{\mathcal{C}}}^1(M,N)$, $(M_0,N_0)\in { \mathrm{Im}}\psi^{\epsilon}$ if and only if where $p$ is the natural projection to $Ext^1_{{\mathcal{C}}}(M,N)$. Furthermore in this case, the fiber of $(M_0,N_0)$ is an affine space and satisfies (ii) For $\eta\in \mathrm{Ext}^1_{{\mathcal{C}}}(N,M)$, $(N_

Theorems & Definitions (48)

  • Lemma 2.1: Chen2021a
  • Lemma 2.2
  • Lemma 2.3: Crawley2002
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: Palu2008
  • Proposition 2.9
  • proof
  • ...and 38 more