Preprojective categories of type A
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911311318745088 |
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| author | Rock, Job Daisie Thomas, Hugh |
| author_facet | Rock, Job Daisie Thomas, Hugh |
| contents | We introduce a continuous version of preprojective algebras of type $A$. In particular, we are interested in the preprojective category over an open, bounded subinterval $\mathbb{I}$ of $\mathbb{R}$, denoted $Λ_{\mathbb{I}}$. We study the representable projective modules and define a useful type of sub- and quotient module called decorous modules. These are completely described by a function from the closure $\overline{\mathbb{I}}$ of $\mathbb{I}$ to $\mathbb{R}$ whose 'slopes' are not too steep anywhere. We later use these to describe permuton ideals, a generalization of the support $τ$-tilting ideals of preprojective algebras of type $A_n$, which we call permutation ideals. Once we have our generalization, we show that permutation ideals can be recovered from permuton ideals. Moreover, permutation ideals are $τ$-rigid and we show an analogous property for our permuton ideals. Along the way, we classify all the brick $Λ_{\mathbb{I}}$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Preprojective categories of type A Rock, Job Daisie Thomas, Hugh Representation Theory We introduce a continuous version of preprojective algebras of type $A$. In particular, we are interested in the preprojective category over an open, bounded subinterval $\mathbb{I}$ of $\mathbb{R}$, denoted $Λ_{\mathbb{I}}$. We study the representable projective modules and define a useful type of sub- and quotient module called decorous modules. These are completely described by a function from the closure $\overline{\mathbb{I}}$ of $\mathbb{I}$ to $\mathbb{R}$ whose 'slopes' are not too steep anywhere. We later use these to describe permuton ideals, a generalization of the support $τ$-tilting ideals of preprojective algebras of type $A_n$, which we call permutation ideals. Once we have our generalization, we show that permutation ideals can be recovered from permuton ideals. Moreover, permutation ideals are $τ$-rigid and we show an analogous property for our permuton ideals. Along the way, we classify all the brick $Λ_{\mathbb{I}}$-modules. |
| title | Preprojective categories of type A |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2512.09618 |