The structure of connected (graded) Hopf algebras

The structure of connected (graded) Hopf algebras revisited

“…By the combinatorial properties of Lyndon words and the standard bracketing on words, one may construct from X a new family of homogeneous generators (z γ ) γ∈N I for H. The coalgebra structure of H assures that this new family of generators satisfy some desirable conditions (Proposition 2.4). Note that several key parts of this observation was already obtained in [24,Proposition 2.4], which is not easy to prove and has its own idea trace back to [12,21]. The decisive new observation for this work is the equality…”

“…In this section, we introduce some basic notations that will be used in the sequel, recall the definition of Lyndon words and present some of their well-known properties for reader's convenience. The materials of Lyndon words are mostly extracted or modified from [7,12,14,24]. Throughout this section, X stands for a well-ordered alphabet.…”

“…Another interesting consequence of Theorem A concerns Hopf Ore extensions. This line of research begun in [18] and has been made great progress recently [5,11,6,24]. Recall that an n-step iterated Hopf Ore extension (n-step IHOE, for short) of a Hopf algebra R is a Hopf algebra H which contains R as a Hopf subalgebra and there is a chain of Hopf subalgebras Note that connected Hopf algebras of finite GK dimension are not necessarilly IHOEs' of the base field.…”

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Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H and a total order on the index set that satisfy several desirable conditions, which reveal some interesting connections between H and K. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].

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Survey on Hopf algebras of GK-dimension 1 and 2

“…Remaining in characteristic 0, but imposing the stronger hypothesis on the Hopf algebra H of finite GK-dimension, that it is connected graded both as an algebra and as a coalgebra, Zhou, Shen and Lu proved the striking result [ZSL19] that H is an IHOE [ZSL19, Theorem B]. Naturally, this leads to Question 3.6.…”