R-Matrix Presentation of Coideal Subalgebra of Quantized Enveloping Algebra of Type Ci

Abstract

The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold. First, for symmetric pairs (sp2n,gln), we construct a coideal subalgebra) of the quantized enveloping algebra of type CI using the R-matrix presentation, based on the work of Noumi. Second, we derive a Poincar´e-Birkhoff-Witt(PBW) basis for) by the A-form approach. As a consequence of the isomorphism btween) and the ıquantum group Uı, our method also yields the PBW basis for the ıquantum group of type CI. Finally, as an application of the R-matrix presentation, we construct a Poisson algebra Pn associated with), and explicitly describe the action of the braid group Bn on the elements of Pn.

Keywords: Quantized enveloping algebra, Coideal subalgebra, Quantum symmetric
pair, Poisson algebra