Bi-genic Hopf algebras in characteristic \(p\)
In this paper, the author computes Ext\(^1(M,M^\prime)\) in the category of modules over the Dieudonné ring \(E\), where \(M,M^\prime\) are Dieudonné modules which correspond to monogenic Hopf algebras \(H,H^\prime\), respectively. Then, the author explicitly describes the Dieudonn\(\acute{e}\) modules that arise as extensions of \(M\) by \(M^\prime\), giving a 2-element generating set. In the case \(M\) and \(M^\prime\) are Dieudonné modules, one can quickly compute Ext\(^1(M,M^\prime)\) using the long exact Hom-Ext sequence, describe the extension group using this technique and show how it agrees with author's classification. Finally, the author determines the algebra structure of a bi-genic Hopf algebra from its Dieudonné module and gives an explicit example of the coalgebra structure for a simple but nontrivial class of Hopf algebras. We hope that future work will examine the lifting questions as in the monogenic case.
scientific article; zbMATH DE number 6152191