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

Abstract

Let $k$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=k\{1,g,h\}$ and its quotient bialgebras (or Hopf algebras) over $k$. We show that the free non-commutative Faà di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha} \mid \alpha(g)\geq 2)$ is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free non-commutative Faà di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative (non-free) version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char$\, k=p>0$, we obtain a chain of quotient Hopf algebras of $\mathcal{T}$ with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.