PBW-bases of coideal subalgebras and a freeness theorem

Abstract

Let H be a character Hopf algebra. Every right coideal subalgebra U that contains the coradical has a PBW-basis which can be extended up to a PBW-basis of H. If additionally U is a bosonization of an invariant with respect to the left adjoint action subalgebra, then H is a free left (and right) U-module with a free PBW-basis over U. These results remain valid if H is a braided Hopf algebra generated by a categorically ordered subset of primitive elements. If the ground field is algebraically closed, the results are still true provided that H is a pointed Hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite-dimensional Yetter-Drinfeld submodules of skew primitive elements.