Coideal Subalgebras and Quantum Symmetric Pairs

TL;DR

This paper surveys coideal subalgebras within quantized enveloping algebras, exploring their generators, modules, and applications to quantum symmetric pairs and spaces, with detailed examples and theoretical insights.

Contribution

It provides a comprehensive study of coideal subalgebras, including new results on their structure, generators, and connections to quantum symmetric pairs and spaces.

Findings

Analysis of generators and modules of coideal subalgebras

Identification of quantum symmetric pairs and their properties

Connections established between coideals and quantum symmetric spaces

Abstract

Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half discusses various well known quantum coideal subalgebras and the implications of the abstract theory on these examples. The focus is on the locally finite part of the quantized enveloping algebra, analogs of enveloping algebras of nilpotent Lie subalgebras, and coideals used to form quantum symmetric pairs. The last family of examples is explored in detail. Connections are made to the construction of quantum symmetric spaces.