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 U-q(g), q(m) not equal 1 containing all group-like elements has a triangular decomposition U = U- circle times(k[F]) k[H] circle times(k[G]) U+, where U- and U+ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if U-1 and U-2 are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular composition U-2 circle times(k[F]) k[H] circle times(k[G]) U-1 is a right coideal but not necessary a subalgebra. Using a recent combinatorial classification of right coideal subalgebras of the quantum Borel algebra U-q(+)(so(2n+1)), we find a necessary condition for the triangular composition to be a right coideal subalgebra of U-q(so(2n+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 u(q)(g). u(q)(so(2n+1)). (C) 2010 Elsevier Inc. All rights reserved.