Classifying submodules over monoidal categories
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910059042177024 |
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| author | Salmasian, Hadi Savage, Alistair Shen, Yaolong |
| author_facet | Salmasian, Hadi Savage, Alistair Shen, Yaolong |
| contents | We study the classification of submodules of module categories over monoidal categories, extending ideas of Coulembier on the classification of tensor ideals in monoidal categories. We develop a framework that applies to module categories equipped with a twisted cylinder twist, a structure closely related to the twisted reflection equation and quantum symmetric pairs. Under mild assumptions, we establish an order-preserving bijection between submodules of a module category $\mathcal{M}$ and submodules of the path-algebra module $\mathcal{M}(1,-)$. We show that this correspondence is compatible with idempotent completion and analyze its behavior under decategorification to the split Grothendieck group, giving criteria for classification in terms of indecomposable objects. As an application, we study the disoriented skein category as a module category over the oriented skein category, describe its indecomposable objects, and obtain a complete classification of its submodules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classifying submodules over monoidal categories Salmasian, Hadi Savage, Alistair Shen, Yaolong Representation Theory Category Theory Quantum Algebra Primary 18M15, 18M30, Secondary 16D90, 17B37 We study the classification of submodules of module categories over monoidal categories, extending ideas of Coulembier on the classification of tensor ideals in monoidal categories. We develop a framework that applies to module categories equipped with a twisted cylinder twist, a structure closely related to the twisted reflection equation and quantum symmetric pairs. Under mild assumptions, we establish an order-preserving bijection between submodules of a module category $\mathcal{M}$ and submodules of the path-algebra module $\mathcal{M}(1,-)$. We show that this correspondence is compatible with idempotent completion and analyze its behavior under decategorification to the split Grothendieck group, giving criteria for classification in terms of indecomposable objects. As an application, we study the disoriented skein category as a module category over the oriented skein category, describe its indecomposable objects, and obtain a complete classification of its submodules. |
| title | Classifying submodules over monoidal categories |
| topic | Representation Theory Category Theory Quantum Algebra Primary 18M15, 18M30, Secondary 16D90, 17B37 |
| url | https://arxiv.org/abs/2603.18264 |