Connected (graded) Hopf algebras

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k of characteristic 0. 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 Hopf 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 = \mathrm {Id}_H, and it 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 algebras 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(\mathfrak{g}) for any Lie algebra \mathfrak{g}.