On the Auslander-Reiten quiver of an infinitesimal group (Q2709967)

The stable Auslander-Reiten quiver for the group algebra \(kG\) of a finite group \(G\) has been studied extensively. In this paper, the author continues investigations of the Auslander-Reiten quiver for the distribution algebra of an infinitesimal group. Using rank varieties for infinitesimal groups and varieties associated to an A-R component of the quiver, he shows that the tree classes of an Auslander-Reiten component are either finite or infinite Dynkin diagrams, or Euclidean diagrams. This is analogous to the determination of tree classes for finite groups by \textit{P. J. Webb} [Math. Z. 179, 97-121 (1982; Zbl 0479.20008)] and extends work of \textit{K. Erdmann} [CMS Conf. Proc. 18, 201-214 (1996; Zbl 0865.16012)] and the author [J. Reine Angew. Math. 464, 47-65 (1995; Zbl 0823.17025)] for restricted Lie algebras or equivalently infinitesimal groups of height \(\leq 1\). The author goes on to find relations between the dimension of the variety of a component and the structure of the component. This is later used to classify the finite and Euclidean components for Frobenius kernels of smooth, reductive groups. Finally, a similar classification is done for supersolvable groups.