Graded pointed Hopf algebras, PBW bases and noncommutative binomial theorem

Abstract

One of pure structure constructions of Hopf algebras is to generate a free Hopf algebra (or a bialgebra) by a coalgebra. Such a free Hopf algebra is usually too big to have nice properties. So we usually consider its quotient Hopf algebras. This construction method produces a large number of examples of pointed Hopf algebras, such as quantum groups.

This PhD thesis consists of two parts: (1) In order to study the quotient Hopf algebras (different from quantum groups) of the pointed free bialgebra $\mathcal{T}$ with PBW bases and GK-dimensions, we use the Lyndon words and the Lyndon-Shirshov basis to study the shuffle type polynomials determining the coproducts of those quotients. In this way, we obtain a free noncommutative binomial theorem, which is strongly related to the well-known Bell polynomials. The related combinatorial results show that the free noncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}$ is an Ore extension of the well-known Fa\`a di Bruno bialgebra. Moreover, a more reasonable noncommutative version of the Fa\`a di Bruno bialgebra is established. Further, we obtain a chain of quotient Hopf algebras of $\overline{\mathcal{T}}$ with finite GK-dimensions over the field of positive characteristic. We study homological properties and the coradical filtrations of those quotients.

Based on the relation of $\mathcal{T}$ and the Fa\`a di Bruno bialgebra, we generalize the Fa\`a di Bruno Hopf algebra and related binomial formula from the point view of pointed Hopf algebras.

(2) Rosso and Kharchenko gave the PBW bases for graded pointed Hopf algebras generated by grouplikes and skew-primitives; Zhou--Shen--Lu gave the PBW bases and structure of graded connected Hopf algebras. Inspired by their work, we use the quantum Lyndon-Shirshov basis to study the PBW bases and structure of graded pointed Hopf algebras of diagonal type not necessarily generated by grouplikes and skew-primitives, which include quantum groups, graded connected Hopf algebras, the generalized Fa\`a di Bruno Hopf algebras, etc..