Let K be a field. We study the free bialgebra T
generated by the coalgebra C=KgβKh and its
quotient bialgebras (or Hopf algebras) over K. We show that the free
noncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of T,
and the quotient bialgebra
T:=T/(EΞ±ββ£Β Ξ±(g)β₯2) is an Ore
extension of the well-known Fa\`a di Bruno bialgebra. The image of the free
noncommutative Fa\`a di Bruno bialgebra in the quotient
T gives a more reasonable non-commutative version of the
commutative Fa\`a di Bruno bialgebra from the PBW basis point view. If char
K=p>0, we obtain a chain of quotient Hopf algebras of
T: Tβ Tnββ Tnβ²β(p)β Tnβ(p)β Tnβ(p;d1β)β β¦β Tnβ(p;djβ,djβ1β,β¦,d1β)β β¦β Tnβ(p;dpβ2β,dpβ3β,β¦,d1β) with finite
GK-dimensions. Furthermore, we study the homological properties and the
coradical filtrations of those quotient Hopf algebras.Comment: 27 pages, typos correcte