Triangular decomposition of right coideal subalgebras

Let g be a Kac–Moody algebra. We show that every homogeneous right coideal subalgebra U of the multiparameter version of the quantized universal enveloping algebra Uq(g), qm≠1 containing all group-like elements has a triangular decomposition U=U−⊗k[F]k[H]⊗k[G]U+, where U− and U+ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if U1 and U2 are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular composition U2⊗k[F]k[H]⊗k[G]U1 is a right coideal but not necessary a subalgebra. Using a recent combinatorial classification of right coideal subalgebras of the quantum Borel algebra , we find a necessary condition for the triangular composition to be a right coideal subalgebra of Uq(so2n+1).If q has a finite multiplicative order t>4, similar results remain valid for homogeneous right coideal subalgebras of the multiparameter version of the small Lusztig quantum groups uq(g), uq(so2n+1).