Universal K-matrix for quantum symmetric pairs

<jats:p>Let <jats:inline-formula id="j_crelle-2016-0012_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1481. png" /> <jats:tex-math>{{\mathfrak{g}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a symmetrizable Kac–Moody algebra and let <jats:inline-formula id="j_crelle-2016-0012_ineq_9998"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1427. png" /> <jats:tex-math>{{U_{q}(\mathfrak{g})}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras <jats:inline-formula id="j_crelle-2016-0012_ineq_9997"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1420. png" /> <jats:tex-math>{{B_{\mathbf{c},\mathbf{s}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula id="j_crelle-2016-0012_ineq_9996"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1427. png" /> <jats:tex-math>{{U_{q}(\mathfrak{g})}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> have a universal K-matrix if <jats:inline-formula id="j_crelle-2016-0012_ineq_9995"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1481. png" /> <jats:tex-math>{{\mathfrak{g}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is of finite type. By a universal K-matrix for <jats:inline-formula id="j_crelle-2016-0012_ineq_9994"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1420. png" /> <jats:tex-math>{{B_{\mathbf{c},\mathbf{s}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> we mean an element in a completion of <jats:inline-formula id="j_crelle-2016-0012_ineq_9993"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1427. png" /> <jats:tex-math>{{U_{q}(\mathfrak{g})}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which commutes with <jats:inline-formula id="j_crelle-2016-0012_ineq_9992"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1420. png" /> <jats:tex-math>{{B_{\mathbf{c},\mathbf{s}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and provides solutions of the reflection equation in all integrable <jats:inline-formula id="j_crelle-2016-0012_ineq_9991"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1427. png" /> <jats:tex-math>{{U_{q}(\mathfrak{g})}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules in category <jats:inline-formula id="j_crelle-2016-0012_ineq_9990"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1470. png" /> <jats:tex-math>{{\mathcal{O}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The construction of the universal K-matrix for <jats:inline-formula id="j_crelle-2016-0012_ineq_9989"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1420. png" /> <jats:tex-math>{{B_{\mathbf{c},\mathbf{s}}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> bears significant resemblance to the construction of the universal R-matrix for <jats:inline-formula id="j_crelle-2016-0012_ineq_9988"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0012_eq_1427. png" /> <jats:tex-math>{{U_{q}(\mathfrak{g})}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. </jats:p> <jats:p>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. </jats:p>