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Triangulated monoidal categorifications of finite type cluster algebras

Élie Casbi

TL;DR

This work develops a triangulated, monoidal categorification framework for finite type cluster algebras by constructing the bounded homotopy category $\mathcal{K}_Q^{(1)}$ from the ambient monoidal category $\mathcal{H}_Q^{(1)}$ tied to a Dynkin quiver $Q$. It introduces a systematic, iterated mapping-cone procedure that yields chain complexes $C_{\bullet}[\beta]$ whose distinguished triangles categorify exchange relations, and proves that their Euler characteristics realize the truncated $q$-characters of simple HL modules, thereby connecting AR-theoretic data to cluster variables. A uniform dominant-monomial formula is established for types $A_n$ and $D_n$, and the construction is positioned to extend to broader cluster structures beyond finite type, illuminating links to higher homological algebra and the broader landscape of monoidal categorifications. The results provide a conceptual bridge between Auslander-Reiten theory, quantum affine representation theory, and cluster algebra combinatorics, with potential implications for interpreting $q$-character coefficients and for developing triangulated categorifications in tame or wild types.

Abstract

We propose a framework of monoidal categorification of finite type cluster algebras involving triangulated monoidal categories. Namely, given a Dynkin quiver $Q$, we consider the bounded homotopy category $\mathcal{K}_Q^{(1)}$ of a symmetric monoidal category $\mathcal{H}_Q^{(1)}$ that we define in terms of the Auslander-Reiten theory of $Q$. Using some iterated mapping cone procedure, we construct a distinguished family $\{ C_{\bullet}[β] \}_{β\in Δ_+}$ of chain complexes in $\mathcal{K}_Q^{(1)}$ characterized (up to isomorphism) by homological conditions similar to those of higher exact sequences appearing in the context of higher homological algebra. We then prove that the distinguished triangle in $\mathcal{K}_Q^{(1)}$ given by each mapping cone categorifies an exchange relation in the finite type cluster algebra $\mathcal{A}_Q$ with initial exchange quiver $Q$ (for a suitable choice of frozen variables). As a consequence, we obtain that for each positive root $β$, the Euler characteristic of $C_{\bullet}[β]$ coincides with the truncated $q$-character of the simple module $L[β]$ in the HL category $\mathcal{C}_ξ^{(1)}$ categorifying the cluster variable $x[β]$ of $\mathcal{A}_Q$ via Hernandez-Leclerc's monoidal categorification. Along the way, we establish a uniform formula for the dominant monomial of $L[β]$ in all types $A_n$ and $D_n$ for arbitrary orientations (agreeing with Brito-Chari's results in type $A_n$).

Triangulated monoidal categorifications of finite type cluster algebras

TL;DR

This work develops a triangulated, monoidal categorification framework for finite type cluster algebras by constructing the bounded homotopy category from the ambient monoidal category tied to a Dynkin quiver . It introduces a systematic, iterated mapping-cone procedure that yields chain complexes whose distinguished triangles categorify exchange relations, and proves that their Euler characteristics realize the truncated -characters of simple HL modules, thereby connecting AR-theoretic data to cluster variables. A uniform dominant-monomial formula is established for types and , and the construction is positioned to extend to broader cluster structures beyond finite type, illuminating links to higher homological algebra and the broader landscape of monoidal categorifications. The results provide a conceptual bridge between Auslander-Reiten theory, quantum affine representation theory, and cluster algebra combinatorics, with potential implications for interpreting -character coefficients and for developing triangulated categorifications in tame or wild types.

Abstract

We propose a framework of monoidal categorification of finite type cluster algebras involving triangulated monoidal categories. Namely, given a Dynkin quiver , we consider the bounded homotopy category of a symmetric monoidal category that we define in terms of the Auslander-Reiten theory of . Using some iterated mapping cone procedure, we construct a distinguished family of chain complexes in characterized (up to isomorphism) by homological conditions similar to those of higher exact sequences appearing in the context of higher homological algebra. We then prove that the distinguished triangle in given by each mapping cone categorifies an exchange relation in the finite type cluster algebra with initial exchange quiver (for a suitable choice of frozen variables). As a consequence, we obtain that for each positive root , the Euler characteristic of coincides with the truncated -character of the simple module in the HL category categorifying the cluster variable of via Hernandez-Leclerc's monoidal categorification. Along the way, we establish a uniform formula for the dominant monomial of in all types and for arbitrary orientations (agreeing with Brito-Chari's results in type ).
Paper Structure (26 sections, 32 theorems, 137 equations)

This paper contains 26 sections, 32 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Auslander-Reiten theory
  3. Auslander-Reiten theoretic background
  4. Additive functions on repetition quivers
  5. Monoidal categorifications
  6. Finite type cluster algebras associated with Dynkin quivers
  7. Representations of affine quantum groups and their $q$-characters
  8. Hernandez-Leclerc categories
  9. Abelian monoidal categorifications
  10. Invariants associated to pairs of simple modules
  11. The underlying category
  12. Quasi-additive functions
  13. Two partial orders on quasi-additive functions
  14. The category $\mathcal{M}_Q$
  15. The category $\mathcal{H}_Q$
  16. ...and 11 more sections

Key Result

Theorem 1.1

For each positive (or negative simple) root $\beta$, there exists a unique (up to isomorphism) strongly $\mathcal{H}_Q^{(1)}$-exact complex $C_{\bullet}[\beta]$ in $\mathcal{K}_Q^{(1)}$ such that $C_n[\beta]=0$ if $n<0$ and $C_0[\beta]$ is indecomposable and isomorphic to $Y[\beta]$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Theorem 3.1: Hernandez-Leclerc HL10, Brito-Chari BC, Kashiwara-Kim-Oh-Park KKOP2
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4: Kashiwara-Kim-Oh-Park
  • Theorem 3.5: Fujita Fujita
  • Definition 4.1
  • Example 4.2
  • ...and 57 more