Coideal subalgebras of pointed and connected Hopf algebras

Abstract

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T 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 R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.