Let \(k\) be an algebraically closed field of characteristic zero. Three new families of Hopf algebras \(T(\mathbf{m},t,\xi)\), \(B(\mathbf{m},\omega,\gamma)\) and \(D(\mathbf{m},d,\gamma)\) are introduced. They depend on the choice of a so-called fraction of a positive integer, a positive integer and a primitive root of unity, satisfying certain conditions (different in each of the three cases). A fraction of length \(\theta\) of a positive integer \(m\) is a theta-tuple \(\mathbf{m}\in {\mathbb N}^\theta\) satisfying the conditions \((e_i,m_i)=1\), \(e_1\cdots e_\theta=m\) and \(\sum_i (a_i-b_i)m_i\not\in m{\mathbb Z}\) if \(\mathbf{a}\neq \mathbf{b}\) in \(A=\{\mathbf{a}\in {\mathcal N}^\theta~:~0\leq a_i<e_i\}\). Here \(e_i=\min\{a~:~m|am_i\}\). These three classes are known in the cases where \(\theta=1\), where we obtain the infinite dimensional Taft algebras , the generalised Liu algebras [K. A. Brown and J. J. Zhang, Proc. Lond. Math. Soc. (3) 101, No. 1, 260–302 (2010; Zbl 1207.16035)] and the algebras \(D(m,d,\gamma)\) introduced in [J. Wu et al., Adv. Math. 296, 1–54 (2016; Zbl 1350.16024)], see Sec. 2 for details.
The algebras \(T(\mathbf{m},t,\xi)\) and \(B(\mathbf{m},\omega,\gamma)\) have GK-dimension one and they are prime. Two algebras belonging to different classes are never isomorphic; two algebras belonging to the same class are isomorphic if certain conditions on the parameters are satisfied.
These examples lead to an improvement of the main result of [J. Wu et al., Adv. Math. 296, 1–54 (2016; Zbl 1350.16024)], in which the classification is given of all affine prime regular Hopf algebras of GK dimension one. Here regularity is replaced by the following two hypothesis:
(Hyp1) \(H\) has a one dimensional representation \(\pi_H:\ H\to k\) whose order is equal to the PI degree of \(H\);
(Hyp2): The invariant components with respect to \(\pi_H\) are domains.
The main result of the paper is Theorem 7.1 stating that a prime Hopf algebra of GK-dimension one satisfying (Hyp1) and (Hyp2) belongs to one of the three families of Hopf algebras \(T(\mathbf{m},t,\xi)\), \(B(\mathbf{m},\omega,\gamma)\) and \(D(\mathbf{m},d,\gamma)\). Several applications, consequences and examples are given throughout the paper.