Coideal subalgebras of pointed and connected Hopf algebras

Abstract:

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.