  We construct a family of right coideal subalgebras of quantum groups, which
have the property that all irreducible representations are one-dimensional, and
which are maximal with this property. The obvious examples for this are the
standard Borel subalgebras expected from Lie theory, but in a quantum group
there are many more. Constructing and classifying them is interesting for
structural reasons, and because they lead to unfamiliar induced (Verma-)modules
for the quantum group. The explicit family we construct in this article
consists of quantum Weyl algebras combined with parts of a standard Borel
subalgebra, and they have a triangular decomposition. Our main result is
proving their Borel subalgebra property. Conversely we prove under some
restrictions a classification result, which characterizes our family. Moreover
we list for Uq(sl4) all possible triangular Borel subalgebras, using our
underlying results and additional by-hand arguments. This gives a good working
example and puts our results into context.
