  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.
