Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic. (Q2354375)

Let \(H\) be a finite dimensional connected Hopf algebra over a field \(k\) of positive characteristic \(p\). Suppose that \(P(H)\) is an abelian restricted Lie algebra of dimension \(d\) and \(\dim H=p^{d+1}\). It is shown that \(H\) is a cocentral extension \(1\to A\to H\to B\to 1\) where \(A,B\) are the restricted universal enveloping algebras of some abelian restricted Lie algebras. So \(H\) is classified up to equivalence by elements of a cohomology group modulo actions of some automorphism group. As an application it is shown that there exists 1-1 correspondence between: (i) the isomorphism classes of semisimple connected Hopf algebras of dimension \(p^{d+1}\) with \(\dim P(H)=d\); (ii) the isomorphism classes of quadratic curves in \(\mathbb P^{d-1}_{\mathbb K}\) if \(p =2\) and of \(\bigwedge(V)\) where \(\dim V=d\) if \(p>2\) up to the affine automorphism group \(\text{PGL}(d,\mathbb K)\); here \(\mathbb K\) is a finite subfield of \(k\); (iii) the isomorphism classes of \(p\)-groups of order \(p^{d+1}\), whose Frattini group is a cyclic group of order \(p\).