Universal K-matrix for quantum symmetric pairs

Abstract

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.