Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911244394430464 |
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| author | Appel, Andrea Vlaar, Bart |
| author_facet | Appel, Andrea Vlaar, Bart |
| contents | Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of trigonometric solutions of Cherednik's generalized reflection equation. These depend upon the choice of a quantum affine symmetric pair $U_q(\mathfrak{k})\subset U_q(\hat{\mathfrak{g}})$. Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs, obtained in our previous work, as well as the fact that every irreducible $U_q(\hat{\mathfrak{g}})$-module is generically irreducible under restriction to $U_q({\mathfrak{k}})$. In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_16503 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras Appel, Andrea Vlaar, Bart Representation Theory Mathematical Physics Quantum Algebra 81R50, 16T25, 17B37, 81R12 Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of trigonometric solutions of Cherednik's generalized reflection equation. These depend upon the choice of a quantum affine symmetric pair $U_q(\mathfrak{k})\subset U_q(\hat{\mathfrak{g}})$. Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs, obtained in our previous work, as well as the fact that every irreducible $U_q(\hat{\mathfrak{g}})$-module is generically irreducible under restriction to $U_q({\mathfrak{k}})$. In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations. |
| title | Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras |
| topic | Representation Theory Mathematical Physics Quantum Algebra 81R50, 16T25, 17B37, 81R12 |
| url | https://arxiv.org/abs/2203.16503 |