Infinitesimal unipotent group schemes of complexity 1 (Q2759161)

Let \(k\) be an algebraically closed field of characteristic \(p\). A \(k\)-group \(\mathcal G\) is uniserial if \(\mathcal G\) has a unique composition series. Uniserial groups play an important role in determining if an infinitesimal group is representation-finite, that is it admits only finitely many isomorphism classes of finite-dimensional indecomposable modules.NEWLINENEWLINENEWLINEThis paper gives a complete classification of isomorphism classes of non-trivial infinitesimal unipotent commutative uniserial \(k\)-groups. It turns out that there are six different types of isomorphism classes, all of which can be described as kernels of Witt vectors (or their duals).NEWLINENEWLINENEWLINEThe classification is facilitated by the use of (classical) Dieudonné modules. A Dieudonné module \(M\) corresponds to a uniserial group if and only if either \(M/FM\) or \(M/VM\) is a simple module (over the Dieudonné ring \(\mathbb{D}\)). In this case, \(M\) is also called uniserial. The authors construct a list of Dieudonné modules with \(M/VM\) simple, and then pass to Cartier duality to get the others.NEWLINENEWLINENEWLINEThe results of this classification enable an examination of representation-finite infinitesimal groups, and the article concludes with such a study, as well as with a discussion of the classification problem for unipotent groups of complexity 1.