Almost split sequences of Verma modules
type: Item
instance of: scholarly article
title: Almost split sequences of Verma modules
zbMATH Open document ID: 1009.16017
author: Rolf Farnsteiner
author: Gerhard Röhrle
published in: Mathematische Annalen
publication date: 2002-06-10T00:00:00Z
review text: 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.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 1753092
zbMATH Keywords: restricted universal enveloping algebras
zbMATH Keywords: almost split sequences
zbMATH Keywords: Dynkin diagrams
zbMATH Keywords: regular components
zbMATH Keywords: restricted Lie algebras
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: indecomposable modules
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: restricted Verma modules
zbMATH Keywords: quasi-simple modules
zbMATH Keywords: Auslander-Reiten components
zbMATH Keywords: tree classes
zbMATH Keywords: support varieties
zbMATH Keywords: simple Lie algebras
MaRDI profile type: Publication
full work available at URL: s002080100279
OpenAlex ID: W1991833863
DOI: 10.1007/S002080100279
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description: scientific article; zbMATH DE number 1753092