RTT presentation of coideal subalgebra of quantized enveloping algebra of type CI

TL;DR

This paper constructs and analyzes a coideal subalgebra of a quantum group of type CI, providing a PBW basis, a Poisson algebra, and braid group actions, advancing the understanding of quantum symmetric pairs.

Contribution

It introduces an $R$-matrix presentation for the coideal subalgebra of type CI and derives a PBW basis, linking it to $ extit{i}$-quantum groups and Poisson structures.

Findings

Constructed a coideal subalgebra $U_q^{tw}(rak{gl}_n)$ using $R$-matrix presentation.

Derived a PBW basis for the subalgebra and the associated $ extit{i}$-quantum group.

Explicitly described the braid group action on the Poisson algebra $rak{P}_n$.

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 Poincar\'e-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 $\imath$quantum group $\mathcal{U}^{\imath}$, our method also yields the PBW basis for the $\imath$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 $\mathcal{P}_n$.