Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

Abstract

Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=\mathbb{K}g\oplus \mathbb{K}h$ and its quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free noncommutative Faà di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha }|\alpha (g)\ge 2)$ is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free noncommutative Faà di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char $\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of $\overline{\mathcal{T}}$: $\overline{\mathcal{T}}\twoheadrightarrow {\overline{\mathcal{T}}}_n\twoheadrightarrow {\overline{\mathcal{T}}}_n^{\prime }(p)\twoheadrightarrow {\overline{\mathcal{T}}}_n(p)\twoheadrightarrow {\overline{\mathcal{T}}}_n(p;d_1)\twoheadrightarrow \dots \twoheadrightarrow {\overline{\mathcal{T}}}_n(p;d_j,d_{j-1},\dots ,d_1)\twoheadrightarrow \dots \twoheadrightarrow {\overline{\mathcal{T}}}_n(p;d_{p-2},d_{p-3},\dots ,d_1)$ with finite GK-dimensions. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

27 pages, typos corrected