Coideal subalgebras of pointed and connected Hopf algebras.

Let H be a pointed Hopf algebra with abelian coradical. Let A\supseteq B be left (or right) coideal subalgebras of H that contain the coradical of H. We show that A has a PBW basis over B, provided that H satisfies certain mild conditions. In the case that H is a connected graded Hopf algebra of characteristic zero and A and B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A is a graded iterated Ore extension of B. These results turn out to be conceptual consequences of a structure theorem for each pair S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S over T. The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don't priorilly assume R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others. [ABSTRACT FROM AUTHOR]