This thesis contains steps towards the classification of all right coideal subalgebras C of a quantum group with generic q, where C has the additional property that all irreducible representations are 1-dimensional and C is maximal with this property. We call such a right coideal subalgebra a Borel subalgebra. First we construct for an arbitrary right coideal
subalgebra a special choice of generators. Then we classify under some restrictions triangular Borelsubalgebras. We give a conjectural description of all Borel subalgebras in the general case, using the description of the graded algebra gr(C). Finally we work out explicitly all Borel subalgebras for Uq(sl2), Uq(sl3) without restrictions and all triangular Borel subalgebras for Uq(sl4).
