RTT 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 $({\mathfrak{sp}}_{2n},{\mathfrak{gl}}_n)$, we construct a coideal subalgebra $U_q^{tw}({\mathfrak{gl}}_n)$ of the quantized enveloping algebra of type CI using the $R$-matrix presentation, based on the work of Noumi. Second, we derive a Poincaré-Birkhoff-Witt(PBW) basis for $U_q^{tw}({\mathfrak{gl}}_n)$ by the $\mathbb{A}$-form approach. As a consequence of the isomorphism btween $U_q^{tw}({\mathfrak{gl}}_n)$ and the $ı$quantum group ${\mathcal{U}}^{ı}$, our method also yields the PBW basis for the $ı$quantum group of type CI. Finally, as an application of the $R$-matrix presentation, we construct a Poisson algebra ${\mathcal{P}}_n$ associated with $U_q^{tw}({\mathfrak{gl}}_n)$, and explicitly describe the action of the braid group ${\mathcal{B}}_n$ on the elements of ${\mathcal{P}}_n$.

33 pages