Universal K-matrix for quantum symmetric pairs
Universal K-matrix for quantum symmetric pairs
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DOI：
10.1515/crelle-2016-0012
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发表时间：
2015-07
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影响因子：
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通讯作者：
M. Balagovic;S. Kolb
中科院分区：
文献类型：
--
作者：
M. Balagovic;S. Kolb
Let {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras {{B_{\mathbf{c},\mathbf{s}}}} of {{U_{q}(\mathfrak{g})}} have a universal K-matrix if {{\mathfrak{g}}} is of finite type. By a universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of {{U_{q}(\mathfrak{g})}} which commutes with {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable {{U_{q}(\mathfrak{g})}} -modules in category {{\mathcal{O}}} . The construction of the universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for {{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.
Let {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras {{B_{\mathbf{c},\mathbf{s}}}} of {{U_{q}(\mathfrak{g})}} have a universal K-matrix if {{\mathfrak{g}}} is of finite type. By a universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of {{U_{q}(\mathfrak{g})}} which commutes with {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable {{U_{q}(\mathfrak{g})}} -modules in category {{\mathcal{O}}} . The construction of the universal K-matrix for {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for {{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.