Almost split sequences of Verma modules
Let \(\mathfrak g\) denote the restricted Lie algebra of a connected reductive algebraic group over an algebraically closed field of prime characteristic \(p>3\). An indecomposable module is called quasi-simple if it has only one predecessor in the Auslander-Reiten quiver. In the paper under review the authors use algebro-geometric methods to show that every restricted Verma module of \(\mathfrak g\) is quasi-simple and belongs to an Auslander-Reiten component of tree class \(A_\infty\). The structure of the Auslander-Reiten components of \(\mathfrak g\) for which the associated support variety is not two-dimensional is well-known by the work of \textit{K. Erdmann} [in: Representation theory of algebras, CMS Conf. Proc. 18, 201-214 (1996; Zbl 0865.16012)] and subsequent work of the first author of the paper under review [J. Reine Angew. Math. 464, 47-65 (1995; Zbl 0823.17025), Indag. Math., New Ser. 10, No. 2, 221-234 (1999; Zbl 1032.17034)]. In the remaining case the authors provide a reduction of the possible location of restricted Verma modules of \(\mathfrak g\) in the Auslander-Reiten quiver to simple Lie algebras of rank two. Here most of the work has to be done and detailed computations involving certain extensions yield the main result stated above.
restricted universal enveloping algebras
scientific article; zbMATH DE number 1753092