The structure of connected (graded) Hopf algebras revisited (Q2675083)

The authors study the structure of connected graded Hopf algebras and they extend and generalize some of the results obtained in [\textit{G. S. Zhou} et al., Adv. Math. 372, Article ID 107292, 30 p. (2020; Zbl 1452.16033)]. In the main theorem of the paper, making use of the combinatorial properties of Lyndon words, the authors show that if \(H\) is a connected graded Hopf algebra and \(K\) is a graded Hopf subalgebra of \(H\), then \(H\) has a homogeneous basis as a left and right graded \(K\)-module with some interesting connections between \(H\) and \(K\). As a consequence of this result, the authors show that every commutative connected (graded) Hopf algebra is a (graded) polynomial algebra over any of its (graded) Hopf subalgebras. Another consequence is related to Hopf Ore extensions, which states that a connected graded Hopf algebra \(H\) of Gelfand-Kirillov dimension \(d\) with a graded Hopf subalgebra \(K\) of Gelfand-Kirillov dimension \(e\) is a graded \((d-e)\)-step iterated Hopf Ore extension of \(K\) of derivation type, which generalizes Theorem B in [loc. cit.].