$H$-based Quivers with potentials and their representations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911173170954240 |
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| author | Lin, Xiaoyue |
| author_facet | Lin, Xiaoyue |
| contents | We generalize Derksen-Weyman-Zelevinsky's theory of quivers with potentials (QPs) to an $H$-based setting by considering quivers with exactly one loop at each vertex, asking the loops to be nilpotent and so attaching a truncated polynomial ring $H_i$ to each vertex. The algebra is then defined by taking the quotient of the complete path algebra by relations arising from analogs of the Jacobian ideals of a given potential. We develop the mutation theory for such $H$-based QPs and their decorated representations in general position. As an application, we consider generalized cluster algebras introduced by Chekhov-Shapiro. For those algebras corresponding to $H$-based quivers $(Q,\mathbf{d})$ that have mutation degree $d_k\leq 2$ at each vertex $k$ and admit nondegenerate potentials $S$ making $(Q,\mathbf{d},S)$ locally free, we provide a representation-theoretic interpretation of $\mathbf{g}$-vectors and $F$-polynomials. When the exchange matrix $B(Q)$ has full rank, we further construct generic character for upper generalized cluster algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $H$-based Quivers with potentials and their representations Lin, Xiaoyue Representation Theory Primary 13F60, Secondary 16G10, 16G20 We generalize Derksen-Weyman-Zelevinsky's theory of quivers with potentials (QPs) to an $H$-based setting by considering quivers with exactly one loop at each vertex, asking the loops to be nilpotent and so attaching a truncated polynomial ring $H_i$ to each vertex. The algebra is then defined by taking the quotient of the complete path algebra by relations arising from analogs of the Jacobian ideals of a given potential. We develop the mutation theory for such $H$-based QPs and their decorated representations in general position. As an application, we consider generalized cluster algebras introduced by Chekhov-Shapiro. For those algebras corresponding to $H$-based quivers $(Q,\mathbf{d})$ that have mutation degree $d_k\leq 2$ at each vertex $k$ and admit nondegenerate potentials $S$ making $(Q,\mathbf{d},S)$ locally free, we provide a representation-theoretic interpretation of $\mathbf{g}$-vectors and $F$-polynomials. When the exchange matrix $B(Q)$ has full rank, we further construct generic character for upper generalized cluster algebras. |
| title | $H$-based Quivers with potentials and their representations |
| topic | Representation Theory Primary 13F60, Secondary 16G10, 16G20 |
| url | https://arxiv.org/abs/2509.19782 |