Pointed Hopf algebras of dimension \(p^3\)
The authors prove that, over an algebraically closed field \(k\) of characteristic 0, if \(H\) is a Hopf algebra with coradical \(kC_p\), \(p\) prime, then the dimension of \(H_1\) is \(3p\). Thus \(H\) is generated by \(c\), a grouplike element of order \(p\), together with skew-primitives \(x\) and \(y\) with \(x^p=y^p=0\) and either \(yx=\lambda xy\) for \(\lambda\) a primitive \(p\)-th root of unity or \(yx=\lambda xy+(c^2-1)\). Combined with results with a similar flavour [\textit{M. Beattie, S. Dăscălescu} and \textit{L. Grünenfelder}, On pointed Hopf algebras of dimension \(p^n\), Proc. Am. Math. Soc. (to appear) and \textit{M. Beattie, S. Dăscălescu} and \textit{L. Grünenfelder}, Constructing pointed Hopf algebras by Ore extensions (preprint)], one sees that there are precisely \(\lfloor(p-1)(p+9)/2\rfloor+5\) isomorphism classes of pointed Hopf algebras of dimension \(p^3\). This classification is also done by different methods [in \textit{N. Andrusekiewitsch} and \textit{H.-J. Schneider}, J. Algebra 209, No. 2, 658-691 (1998) and \textit{D. Stefan} and \textit{F. Van Oystaeyen}, Applications, J. Algebra 210, No. 2, 535-556 (1998)].
scientific article; zbMATH DE number 1247619