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Fusion-stable structures on triangulated categories

Yu Qiu, Xiaoting Zhang

TL;DR

This work extends tilting, mutation, and stability theory to a fusion-stable setting by allowing a fusion category $\\mathcal{G}$ to act on triangulated categories, and proves a deformation theorem for fusion-stable stability conditions. It then develops fusion-stable tilting theory, exchange graphs, and stability spaces, and shows how fusion-weighted folding provides a uniform way to realize cluster-exchange graphs and hyperplane arrangements in ADE and non-simply laced types. The two main applications show that cluster combinatorics of finite types can be categorified as fusion-stable structures, and that the universal cover of Coxeter-Dynkin hyperplane arrangements arises from the space of fusion-stable stability conditions, yielding a new uniform proof of the $K(\\pi,1)$-conjecture in finite type. Overall, the paper provides a cohesive framework unifying folding, fusion actions, and stability phenomena across quivers, Calabi–Yau completions, and 2-Calabi–Yau categories, with potential impacts on categorification, representation theory, and geometric group theory.

Abstract

Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's construction. We study $\mathcal{G}$-stable tilting, cluster and stability structures on $\mathcal{D}$. In particular, we prove the deformation theorem for $\mathcal{G}$-stable stability conditions. A first application is that Duffield-Tumarkin's categorification of cluster exchange graphs of finite Coxeter-Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter-Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of $K(π,1)$-conjecture in the finite Coxeter-Dynkin case.

Fusion-stable structures on triangulated categories

TL;DR

This work extends tilting, mutation, and stability theory to a fusion-stable setting by allowing a fusion category to act on triangulated categories, and proves a deformation theorem for fusion-stable stability conditions. It then develops fusion-stable tilting theory, exchange graphs, and stability spaces, and shows how fusion-weighted folding provides a uniform way to realize cluster-exchange graphs and hyperplane arrangements in ADE and non-simply laced types. The two main applications show that cluster combinatorics of finite types can be categorified as fusion-stable structures, and that the universal cover of Coxeter-Dynkin hyperplane arrangements arises from the space of fusion-stable stability conditions, yielding a new uniform proof of the -conjecture in finite type. Overall, the paper provides a cohesive framework unifying folding, fusion actions, and stability phenomena across quivers, Calabi–Yau completions, and 2-Calabi–Yau categories, with potential impacts on categorification, representation theory, and geometric group theory.

Abstract

Let be a fusion category acting on a triangulated category , in the sense that is a -module category. Our motivation example is fusion-weighted species, which is essentially Heng's construction. We study -stable tilting, cluster and stability structures on . In particular, we prove the deformation theorem for -stable stability conditions. A first application is that Duffield-Tumarkin's categorification of cluster exchange graphs of finite Coxeter-Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter-Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of -conjecture in the finite Coxeter-Dynkin case.
Paper Structure (32 sections, 9 theorems, 74 equations, 4 figures)

This paper contains 32 sections, 9 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introductions
  2. Preliminaries
  3. Basics
  4. Additive categories
  5. Coxeter diagram
  6. Braid and Weyl groups
  7. Various structures
  8. Torsion pairs, t-structures and hearts
  9. Silting/cluster tilting
  10. Tilting and mutation
  11. Stability structure
  12. The quiver categories
  13. Fusion actions on triangulated categories
  14. Fusion categories and actions
  15. Fusion actions
  16. ...and 17 more sections

Key Result

Lemma 4.5

Let $\mu^\sharp_{\mathcal{G}(S)}\colon\mathcal{H}\rightarrow\mathcal{H}^\sharp_{\mathcal{G}(S)}$ be a $\mathcal{G}$-simple forward tilting. If $\mathcal{G}(S)$ is rigid (i.e. $\operatorname{Ext}^1$-vanishing) and the indecomposable objects in $\mathcal{G}(S)$ are simples $S_1,\ldots,S_l$ in $\mathca

Figures (4)

  • Figure 1: The cluster exchange graphs $\underline{\operatorname{CEG}}(\Delta)$ for type $A_3/B_3/H_3$. Type $B_3/H_3$ can be obtained by weighted folding type $D_4/D_6$, cf. \ref{['fig:BCEI']} and \ref{['fig:H234']}, respectively.
  • Figure 2: A complete list of finite Coxeter graphs
  • Figure 3: Weighted folding for type $B_{\mathbf{n}}/C_{\mathbf{n}}$ and $F_4$
  • Figure 4: Weighted folding type $H_3/H_4$ and $I_2(h)$

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5: Folding of finite type quivers
  • Definition 3.6
  • ...and 28 more