Classification of affine prime regular Hopf algebras of GK-dimension one. (Q277978)

Throughout the review, all Hopf algebras considered are over an algebraically closed field of characteristic zero. Additionally, all considered Hopf algebras are assumed to be affine and Noetherian. In [Proc. Lond. Math. Soc. (3) 101, No. 1, 260--302 (2010; Zbl 1207.16035)], \textit{K. A. Brown} and \textit{J. J. Zhang} classified all prime regular Hopf algebras \(H\) of Gel'fand-Kirillov dimension one under the hypothesis that \(\mathrm{im}(H)=1\) or \(\mathrm{im}(H)=\mathrm{io}(H)\), where \(\mathrm{im}(H)\) and \(\mathrm{io}(H)\) mean the integral minor of \(H\) and the integral order of \(H\), respectively, and raised the open question of whether the Brown and Zhang classification remains valid when the hypothesis that \(\mathrm{im}(H)\) is one or \(\mathrm{io}(H)\), is dropped.NEWLINENEWLINE In the paper under review, the authors continue Brown and Zhang's work, and finish the classification of all prime regular Hopf algebras of GK-dimension one. The authors construct a new class of prime regular Hopf algebras \(D(m,d,\xi)\) of GK-dimension one with \(\mathrm{im}(D(m,d,\xi))=m\) and \(\mathrm{io}(D(m,d,\xi))=2m\). Since \(D(m,d,\xi)\) is not a pointed Hopf algebra, it follows that \(D(m,d,\xi)\) is not isomorphic to any one of Hopf algebras mentioned by Brown and Zhang [loc. cit.]. The authors of the paper under review prove that their new class together with the four classes of Hopf algebras mentioned by Brown and Zhang form a complete list, up to Hopf algebra isomorphisms, of all prime regular Hopf algebras of GK-dimension one.