  We study algebraic and homological properties of two classes of infinite
dimensional Hopf algebras over an algebraically closed field k of
characteristic zero. The first class consists of those Hopf k-algebras that are
connected graded as algebras, and the second class are those Hopf k-algebras
that are connected as coalgebras. For many but not all of the results presented
here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension.
It is shown that if the Hopf algebra H is a connected graded algebra of finite
Gel'fand-Kirillov dimension n, then H is a noetherian domain which is
Cohen-Macaulay, Artin-Schelter regular and Auslander regular of global
dimension n. It has S^2 = Id_H, and is Calabi-Yau. Detailed information is also
provided about the Hilbert series of H. Our results leave open the possibility
that the first class of algebras is (properly) contained in the second. For
this second class, the Hopf k-algebras of finite Gel'fand-Kirillov dimension n
with connected coalgebra, the underlying coalgebra is shown to be
Artin-Schelter regular of global dimension n. Both these classes of Hopf
algebra share many features in common with enveloping algebras of finite
dimensional Lie algebras. For example, an algebra in either of these classes
satisfies a polynomial identity only if it is a commutative polynomial algebra.
Nevertheless, we construct, as one of our main results, an example of a Hopf
k-algebra H of Gel'fand-Kirillov dimension 5, which is connected graded as an
algebra and connected as a coalgebra, but is not isomorphic as an algebra to
U(g) for any Lie algebra g.
