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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.25012 |
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| _version_ | 1866912622199177216 |
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| author | Dequêne, Benjamin Roy, Sunny |
| author_facet | Dequêne, Benjamin Roy, Sunny |
| contents | On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties.
On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data.
We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting object $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras Dequêne, Benjamin Roy, Sunny Representation Theory Combinatorics On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting object $T$. |
| title | Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2509.25012 |