Left coideal subalgebras of Nichols algebras

We determine all Nichols algebras of finite-dimensional Yetter-Drinfeld modules over groups such that all its left coideal subalgebras in the category of ${\mathbb{N}}_0$-graded comodules over the group algebra are generated in degree one as an algebra. Here we confine ourselves to Yetter-Drinfeld modules in which each group-homogeneous component is at most one-dimensional. We present a strategy to extend left coideal subalgebras by adding a suitable generator in degree two, three or four to a smaller left coideal subalgebra. We also discuss some methods for the construction of left coideal subalgebras of a Nichols algebra in the category of ${\mathbb{N}}_0$-graded $H$-comodules, where $H$ is a Hopf algebra, that is not necessarily a group algebra.