On cocommutative Hopf algebras of finite representation type
Let \(\mathcal G\) be a finite algebraic group over an algebraically closed field \(k\) of positive characteristic. The structure of the ``infinitesimal part'' (i.e. the largest normal infinitesimal subgroup \({\mathcal G}^0\subset{\mathcal G}\)) is determined up to the classification of what are called ``\(V\)-uniserial groups''. A group \(\mathcal U\) is called \(V\)-uniserial if the Verschiebung \(V_{\mathcal U}\) induces an exact sequence \({\mathcal U}^{(p)}\to{\mathcal U}\to{_p\alpha_k}\to e_k\). The term \(V\)-uniserial is used because in this case \(\mathcal U\) has exactly one composition series as its subgroups are of the form \(V_{\mathcal U}^i({\mathcal U}^{(p^i)})\). Initially, the paper focuses on infinitesimal groups of finite representation type. It is shown that such a group is supersolvable, and this observation leads to a description of the structure of such groups. The principal blocks are representation-finite, as is shown in Theorem 2.7. The classification is shown to reduce to the classification of \(V\)-uniserial groups, however the classification of these groups does not appear here -- the authors state that its solution ``involves different techniques and is therefore relegated to a separate paper.'' Indeed, the calculations here are homological, whereas it is stated that future papers will rely more on geometry. The topic is then generalized to dimension zero algebraic groups that are representation-finite. It is shown that such a Hopf algebra is representation-finite if and only if the constant and infinitesimal components are representation-finite and at least one of the components is semi-simple. Finally, it is shown how some of the results are related to work by \textit{A. Premet} [Math. Z. 228, No. 2, 255-282 (1998; Zbl 0913.17011)].
scientific article; zbMATH DE number 1545684
scientific article