Kohomologie von p-Lie-Algebren und nilpotente Elemente. (Cohomology of p- Lie-algebras and nilpotent elements)
type: Item
instance of: scholarly article
title: Kohomologie von p-Lie-Algebren und nilpotente Elemente. (Cohomology of p- Lie-algebras and nilpotent elements)
zbMATH Open document ID: 0614.17008
DOI: 10.1007/BF02941516
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a finite dimensional p-Lie algebra over an algebraically closed field k of characteristic \(p>0\). If M is a restricted module for \({\mathfrak g}\), then \(H_*({\mathfrak g},M)\) denotes the restricted cohomology of M. Then the variety attached to M is the variety \(V_{{\mathfrak g}}(M)\) defined by the annihilator in S(\({\mathfrak g}^*)\) of H(\({\mathfrak g},M\otimes M^*)\). It is first proved that \(V_{{\mathfrak g}}(k)\) consists of those elements in \({\mathfrak g}\) which are killed by the p-th power map and next that if either \(p=2\) or if \({\mathfrak g}\) is unipotent, then \(V_{{\mathfrak g}}(M)=\{x\in V_{{\mathfrak g}}(k)|\) M's restriction to kx is not injective\(\}\cup \{0\}\). The last statement was soon after proved for all \({\mathfrak g}\) by \textit{E. M. Friedlander} and \textit{B. J. Parshall} [Invent. Math. 86, 553-562 (1986)].
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 3993812
zbMATH Keywords: restricted Lie algebra
zbMATH Keywords: support variety
zbMATH Keywords: restricted cohomology
author: Jens Carsten Jantzen
reviewed by: Henning Haahr Andersen
MaRDI profile type: Publication
cites work: Cohomology of induced representations for algebraic groups
cites work: Q5580319
cites work: Q5602064
cites work: Rational representations of algebraic groups: Tensor products and filtrations
cites work: On the cohomology of algebraic and related finite groups
cites work: Cohomology of Lie Algebras and Algebraic Groups
cites work: Cohomology of infinitesimal and discrete groups
cites work: Cohomology of Restricted Lie Algebras
cites work: Q4775998
cites work: Lie Group Representations on Polynomial Rings
cites work: Some arithmetical results on semi-simple Lie algebras
cites work: Conjugacy classes in algebraic groups. Notes by Vinay V. Deodhar
full work available at URL: https://doi.org/10.1007/bf02941516
OpenAlex ID: W2038498061
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description: scientific article; zbMATH DE number 3993812