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

Yingwen Zhang, Hongda Lin, Honglian Zhang

TL;DR

The paper addresses constructing and analyzing a CI-type coideal subalgebra inside the quantized enveloping algebra $U_q(\mathfrak{sp}_{2n})$ using RTT-type relations and the reflection equation. It establishes a PBW basis via the $\mathbb{A}$-form, proves an explicit isomorphism to the $\imath$quantum group $\mathcal{U}^{\imath}$ of CI type, and constructs a Poisson algebra $\mathcal{P}_n$ as the $q\to 1$ limit with an explicit braid-group action. The results unify Noumi–Letzter twisted algebras with Letzter–Kolb quantum symmetric pairs in type CI, and provide a concrete bridge between quantum and classical (Poisson) structures, including a detailed description of the braid-group symmetries on both the algebra and Poisson sides.

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 $\imath$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 $\mathcal{P}_n$.

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

TL;DR

The paper addresses constructing and analyzing a CI-type coideal subalgebra inside the quantized enveloping algebra using RTT-type relations and the reflection equation. It establishes a PBW basis via the -form, proves an explicit isomorphism to the quantum group of CI type, and constructs a Poisson algebra as the limit with an explicit braid-group action. The results unify Noumi–Letzter twisted algebras with Letzter–Kolb quantum symmetric pairs in type CI, and provide a concrete bridge between quantum and classical (Poisson) structures, including a detailed description of the braid-group symmetries on both the algebra and Poisson sides.

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 , we construct a coideal subalgebra of the quantized enveloping algebra of type CI using the -matrix presentation, based on the work of Noumi. Second, we derive a Poincaré-Birkhoff-Witt(PBW) basis for by the -form approach. As a consequence of the isomorphism btween and the quantum group , our method also yields the PBW basis for the quantum group of type CI. Finally, as an application of the -matrix presentation, we construct a Poisson algebra associated with , and explicitly describe the action of the braid group on the elements of .
Paper Structure (9 sections, 24 theorems, 150 equations)

This paper contains 9 sections, 24 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. The quantized enveloping algebra for type $\mathfrak{sp}_{2n}$
  3. Coideal subalgebra of quantized enveloping algebra for type $U_q(\mathfrak{sp}_{2n})$
  4. Symmetric pair $(\mathfrak{sp}_{2n},\mathfrak{gl}_n$)
  5. Coideal subalgebra $U_q^{tw}(\mathfrak{gl}_n)$ and its PBW basis
  6. An equivalent presentation of $U_q^{tw}(\mathfrak{gl}_n)$
  7. The proof of the theorem \ref{['pbw']}
  8. Connection with $\imath$quantum group
  9. An associated Poisson algebra

Key Result

Proposition 2.4

([NL90,Theorem 12.]) There exists a Hopf algebraic isomorphism ${\phi}$ from $U_q(\mathfrak{sp}_{2n})$ to $\mathcal{U}_q(\mathfrak{sp}_{2n})$ such that where $k_i=t_it_{i+1}^{-1}$, for $1\leq i\leq n$.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 48 more