2-categorical affine symmetries of quantum enveloping algebras
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908964449419264 |
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| author | Qunell, Sam |
| author_facet | Qunell, Sam |
| contents | We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of $U_q^+(\mathfrak{sl}_{n+1})$ on itself and are closely related to evaluation morphisms of quantum groups. We expect that our 2-representation exists in all simple types and show that the corresponding 1-representation exists in types $D_4$ and $C_2$. We also show that a certain quotient of our 1-representation in type $A_n$ is isomorphic to a prefundamental representation. We use this to provide a new proof of the prefundamental representation character formulas in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2-categorical affine symmetries of quantum enveloping algebras Qunell, Sam Quantum Algebra Representation Theory 81R50 (Primary) 17B37 (Secondary) We produce 2-representations of the positive part of affine quantum enveloping algebras on their finite-dimensional counterparts in type $A_n$. These 2-representations naturally extend the right-multiplication 2-representation of $U_q^+(\mathfrak{sl}_{n+1})$ on itself and are closely related to evaluation morphisms of quantum groups. We expect that our 2-representation exists in all simple types and show that the corresponding 1-representation exists in types $D_4$ and $C_2$. We also show that a certain quotient of our 1-representation in type $A_n$ is isomorphic to a prefundamental representation. We use this to provide a new proof of the prefundamental representation character formulas in these cases. |
| title | 2-categorical affine symmetries of quantum enveloping algebras |
| topic | Quantum Algebra Representation Theory 81R50 (Primary) 17B37 (Secondary) |
| url | https://arxiv.org/abs/2502.08039 |