Extensions of simple modules for quantum groups at complex roots of $1$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913996273090560 |
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| author | Andersen, Henning Haahr |
| author_facet | Andersen, Henning Haahr |
| contents | Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ is a root of unity. In this paper we study the extensions between simple modules in the category consisting of the finite dimensional modules for $U_q$. We first prove that this problem is equivalent to finding the extensions between the finitely many simple modules for the small quantum group $u_q$ in $U_q$. Then we show that the extension groups in question are determined by a finite subset with small highest weights. When the order of $q^2$ is at least the Coxeter number for $R$ we prove that the dimensions of such extension groups equal the top degree coefficients of some associated Kazhdan-Lusztig polynomial for the affine Weyl group for $R$.
We relate all this to similar (old) results for almost simple algebraic groups and their Frobenius subgroup schemes over fields of large prime characteristics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensions of simple modules for quantum groups at complex roots of $1$ Andersen, Henning Haahr Representation Theory 20G42 Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ is a root of unity. In this paper we study the extensions between simple modules in the category consisting of the finite dimensional modules for $U_q$. We first prove that this problem is equivalent to finding the extensions between the finitely many simple modules for the small quantum group $u_q$ in $U_q$. Then we show that the extension groups in question are determined by a finite subset with small highest weights. When the order of $q^2$ is at least the Coxeter number for $R$ we prove that the dimensions of such extension groups equal the top degree coefficients of some associated Kazhdan-Lusztig polynomial for the affine Weyl group for $R$. We relate all this to similar (old) results for almost simple algebraic groups and their Frobenius subgroup schemes over fields of large prime characteristics. |
| title | Extensions of simple modules for quantum groups at complex roots of $1$ |
| topic | Representation Theory 20G42 |
| url | https://arxiv.org/abs/2508.12898 |