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- Title
Universal K-matrix for quantum symmetric pairs.
- Authors
Balagović, Martina; Kolb, Stefan
- Abstract
Let 𝔤 {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let U q (𝔤) {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} of U q (𝔤) {{U_{q}(\mathfrak{g})}} have a universal K-matrix if 𝔤 {{\mathfrak{g}}} is of finite type. By a universal K-matrix for B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of U q (𝔤) {{U_{q}(\mathfrak{g})}} which commutes with B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable U q (𝔤) {{U_{q}(\mathfrak{g})}} -modules in category 𝒪 {{\mathcal{O}}}. The construction of the universal K-matrix for B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for U q (𝔤) {{U_{q}(\mathfrak{g})}}. Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.
- Subjects
RIBBONS; CYLINDER (Shapes); QUANTUM chemistry; MATRICES (Mathematics); DIECK, Tom
- Publication
Journal für die Reine und Angewandte Mathematik, 2019, Vol 2019, Issue 747, p299
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2016-0012