3150 2010 eng 1 33 33 preprint 0 -- 2025-12-18 -- We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. We apply the criterion to classical examples and liftings of Nichols algebras which were determined in arxiv.org:1003.5882. 10.48550/arxiv.1011.0140 Nein publish false false 2 Michael Helbig eng uncontrolled Quantum Algebra eng uncontrolled Hopf algebra eng uncontrolled Nichols algebra Mathematik Fakultät für Angewandte Natur- und Geisteswissenschaften Technische Hochschule Rosenheim