The Hopf Algebra of a Uniserial Group

Let k be an algebraically closed field of characteristic p > 0. The purpose of this paper is to describe the Hopf algebra of a finite commutative infinitesimal unipotent k-group scheme which is uniserial, i.e., which has a unique composition series. As there is only one simple finite commutative infinitesimal unipotent group scheme (namely αp := ker {F : Ga → Ga} , with Ga being the additive group scheme and F the Frobenius map), composition series on this class of group schemes are a bit easier to study than for arbitrary group schemes. A certain class of uniserial groups, namely the V -uniserial groups, are important in studying representation theory: A finite connected k-group scheme G has finite representation type if and only if the quotient G/M(G), where M(G) is the multiplicative center of G, is a semidirect product of a V -uniserial unipotent group U together with a group of type μpn [FV, 2.7]. In [FRV], the authors introduce Dieudonné modules to classify the V uniserial unipotent groups in an effort to describe all groups of finite representation type. It is shown that the (isomorphism classes of) uniserial groups follow one of six different “types”, three of which are dual to the other three. Here, we will also use classical Dieudonné module theory to describe uniserial groups, but we will reduce the number of types needed in the description. Surprisingly, there is an easy way to write the isomorphism classes of uniserial groups in terms of Dieudonné modules: They all fit into one of two types, and furthermore the types are dual to each other. Since any representation-finite local algebra is of the form k [t] / (tm), by [FV, 2.7 and 3.1] we have that G = Spec (H) is V -uniserial if and only if H∗ = Hom(H, k) is monogenic (i.e., generated as a k-algebra by a single element), and by duality G is F -uniserial if and only if H is monogenic. Using the classification