Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension (Q2840172)

The Gelfand-Kirillov dimension (or GK-dimension for short) has been a useful tool for studying and investigating infinite-dimensional Hopf algebras. An interesting phenomenon is that the GK-dimension of every known Hopf algebra is either infinity or a non-negative integer. So, the author in this paper is tempting to conjecture that this is always true for any Hopf algebra. As positive evidence for this conjecture, he proves that the GK-dimension of a connected Hopf algebra over an algebraically closed field of characteristic 0 is either infinity or a positive integer. Assume that the base field \(k\) is algebraically closed of characteristic 0 and \(H\) is a connected Hopf algebra. Then the following statements are equivalent:NEWLINENEWLINE {\parindent=6mm \begin{itemize}\item[(1)] GKdim \(H<\infty;\) \item[(2)] GKdim gr\(H<\infty\); \item[(3)] gr\(H\) is finitely generated; (4) gr\(H\) is isomorphic to the polynomial ring of \(\ell\) variables for some \(\ell\geq 0\) as algebras. In this case, GKdim\(H=\)GKdim gr\(H\), which is a positive integer. NEWLINENEWLINE\end{itemize}} Moreover, he classifies connected Hopf algebras of GK-dimension 3 over an algebraically closed field of characteristic 0.