\(R\)-Hopf algebra orders in \(KC_{p^ 2}\)
Let \(K\) be a finite extension field of the \(p\)-adic rationals and let \(R\) be the integral closure of the \(p\)-adic integers in \(K\). For any finite group \(G\), the group algebra \(KG\) is a Hopf algebra, and an \(R\)-order in \(KG\) is said to be a Hopf order if it inherits a Hopf algebra structure from \(KG\). In the case that \(G = C_ p\) is cyclic of order \(p\), Larson has described all the \(R\)-Hopf orders in \(KC_ p\), and this construction was extended by Greither to obtain some types of \(R\)-Hopf order in \(KC_{p^ 2}\). In the present paper, the author uses Larson's results, together with ingenious computations, to complete the description of \(R\)- orders in \(KC_{p^ 2}\): any such order is either a Greither order or the linear dual of a Greither order.
scientific article; zbMATH DE number 682580