A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912857976733696 |
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| author | Grantcharov, Dimitar Nguyen, Khoa Zhao, Kaiming |
| author_facet | Grantcharov, Dimitar Nguyen, Khoa Zhao, Kaiming |
| contents | We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21197 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$ Grantcharov, Dimitar Nguyen, Khoa Zhao, Kaiming Representation Theory 17B10, 17B20 We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form. |
| title | A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$ |
| topic | Representation Theory 17B10, 17B20 |
| url | https://arxiv.org/abs/2601.21197 |