Extension functors of modular Lie algebras
type: Item
instance of: scholarly article
title: Extension functors of modular Lie algebras
zbMATH Open document ID: 0731.17007
DOI: 10.1007/BF01444560
author: Rolf Farnsteiner
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164762
review text: One often observes fundamental differences between the cohomological aspects of modular and nonmodular representation theory of Lie algebras. For instance, the fact that for any nonzero modular Lie algebra L, \(H^ 1(L,M)\neq 0\) for a suitably chosen finite L-module M contrasts markedly with Weyl's classical theorem. The author first introduces the notion of a generalized reduced Verma module which unifies various induction functors hitherto considered. The functors \(Ext^ n(M\),-) and \(Ext^ n(\)-,M) with values in generalized reduced Verma modules are investigated. Using the product theory of extension functors, several reduction theorems are established which elicit the conceptual source of the structural peculiarities of modular cohomology theory. Nonvanishing results are obtained which ensure the existence of finite dimensional modular representations \(\rho\) : \(L\to gl(N)\) such that \(Ext^ i(M,N)\neq 0\neq Ext^ i(N,M)\), \(0\leq i\leq \dim L\). Finally, a description of the ordinary and relative cohomology of a \({\mathbb{Z}}\)-graded Lie algebra L with coefficients in generalized reduced Verma modules is given.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 4208329
zbMATH Keywords: modular Lie algebra
zbMATH Keywords: generalized reduced Verma module
zbMATH Keywords: extension functors
zbMATH Keywords: reduction theorems
zbMATH Keywords: Nonvanishing
zbMATH Keywords: modular representations
zbMATH Keywords: cohomology
reviewed by: Guangyu Shen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208329