Triangulated monoidal categorifications of finite type cluster algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912853353562112 |
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| author | Casbi, Élie |
| author_facet | Casbi, Élie |
| contents | 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 $A_n$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19754 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Triangulated monoidal categorifications of finite type cluster algebras Casbi, Élie Representation Theory 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 $A_n$). |
| title | Triangulated monoidal categorifications of finite type cluster algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2601.19754 |