The main aim of the paper under review is to classify all connected Hopf algebras of dimension \(p^2\) over an algebraically closed field \(\mathbf k\) of positive characteristic \(p\). Let \(H\) be such a Hopf algebra with the primitive space \(P(H)\). According to Proposition 2.2, \(\dim P(H)\leq 2\). The author considers both cases separately.
He proves in Theorem 7.4 that if \(\dim P(H)=2\), then \(H\) is isomorphic to one of the following: (1) \(\mathbf k[x,y]/(x^p,y^p)\); (2) \(\mathbf k[x,y]/(x^p-x,y^p)\); (3) \(\mathbf k[x,y]/(x^p-y,y^p)\); (4) \(\mathbf k[x,y]/(x^p-x,y^p-y)\); (5) \(\mathbf k[x,y]/([x,y]-y,x^p-x,y^p)\), where \(x\) and \(y\) are primitive.
If \(\dim P(H)=1\), then \(H\) is isomorphic to one of the following: (6) \(\mathbf k[x,y]/(x^p,y^p)\); (7) \(\mathbf k[x,y]/(x^p,y^p-x)\); (8) \(\mathbf k[x,y]/(x^p-x,y^p-y)\), where the coalgebra structure is given by \(\Delta(x)=x\otimes 1+1\otimes x\) and \(\Delta(y)=y\otimes 1+1\otimes y+\sum_{i=1}^{p-1}\frac{(p-1)!}{i!\cdot(p-i)!}x^i\otimes x^{p-i}\).
Finally, the author classifies all local Hopf algebras of dimension \(p^2\) over \(\mathbf k\) (Corollary 7.5).