  For a quantum group, we study those right coideal subalgebras, for which all
irreducible representations are one-dimensional. If a right coideal subalgebra
is maximal with this property, then we call it a Borel subalgebra.
  Besides the positive part of the quantum group and its reflections, we find
new unfamiliar Borel subalgebras, for example, ones containing copies of the
quantum Weyl algebra. Given a Borel subalgebra, we study its induced
(Verma-)modules and prove among others that they have all irreducible
finite-dimensional modules as quotients. We give two structural conjectures
involving the associated graded right coideal subalgebra, which we prove in
certain cases. In particular, they predict the shape of all triangular Borel
subalgebras. As examples, we determine all Borel subalgebras of
$U_q(\mathfrak{sl}_2)$ and $U_q(\mathfrak{sl}_3)$ and discuss the induced
modules.
