I will report on recent joint work with I. Heckenberger. We
are studying systematically the Nichols algebra (or quantum symmetric
algebra) of a Yetter-Drinfeld module over any Hopf algebra (with
bijective antipode) which is a finite direct sum of finite-dimensional
irreducible Yetter-Drinfeld modules. In this general context we define
reflection maps in joint work with N. Andruskiewistch. In the special
case of the quantum groups in Lusztig's book these maps are
essentially the restriction of the Lusztig automorphisms to the plus
part of the quantum group. Under mild assumptions we associate a
generalized root system (in the sense of Heckenberger and Yamane) and
a Weyl groupoid to the Nichols algebra. Using these invariants it is
possible to decide when the Nichols algebra is finite-dimensional. We
obtain a coproduct formula which seems to be new even for the
classical quantum groups. Then we describe the right coideal
subalgebras of the Nichols algebra by words in the Weyl groupoid. As a
special case we obtain a proof of a recent conjecture of Kharchenko
which says that the number of right coideal subalgebras of the plus
part of the quantum group of a semisimple Lie algebra is the order of
the Weyl group.