Universal K-matrices for quantum Kac-Moody algebras
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911222079684608 |
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| author | Appel, Andrea Vlaar, Bart |
| author_facet | Appel, Andrea Vlaar, Bart |
| contents | We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra $U_qL\mathfrak{sl}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_09218 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Universal K-matrices for quantum Kac-Moody algebras Appel, Andrea Vlaar, Bart Representation Theory Mathematical Physics Quantum Algebra 81R10, 17B37, 17B67, 16T10 We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra $U_qL\mathfrak{sl}_2$. |
| title | Universal K-matrices for quantum Kac-Moody algebras |
| topic | Representation Theory Mathematical Physics Quantum Algebra 81R10, 17B37, 17B67, 16T10 |
| url | https://arxiv.org/abs/2007.09218 |