On right coideal subalgebras of quantum groups

TL;DR

This paper studies right coideal subalgebras within quantum groups, providing a classification of their generators and explicitly determining these subalgebras for Uq(sl2) and Uq(sl3), with implications for their representations.

Contribution

It proves that any right coideal subalgebra has a well-structured set of generators, facilitating their classification and explicit determination in specific quantum groups.

Findings

Right coideal subalgebras have a nice generating set.

Explicit classification of right coideal subalgebras for Uq(sl2) and Uq(sl3).

Discussion of their representation theoretic properties.

Abstract

Right coideal subalgebras are interesting substructures of Hopf algebras such as quantum groups. Examples of right coideal subalgebras are the quantum Borel part as well as quantum symmetric pairs. Classifying right coideal subalgebras is a difficult question with notable results by Schneider, Heckenberger and Kolb. After reviewing these results, as main result we prove that an arbitrary right coideal subalgebras has a particularly nice set of generators. This allows in principle to specify the set of right coideal subalgebras in a given case. As application we determine right coideal subalgebras of the quantum groups Uq(sl2) and Uq(sl3) and discuss their representation theoretic properties.