On the cohomology of a new class of Lie algebras generalizing Borel subalgebras
type: Item
instance of: scholarly article
title: On the cohomology of a new class of Lie algebras generalizing Borel subalgebras
zbMATH Open document ID: 0796.17019
author: Roger Carles
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: It is well known that the second cohomology groups of Lie algebras in an adjoint module describe its infinitesimal deformations. The condition \(H^ 2({\mathfrak g},{\mathfrak g})=0\) means that \({\mathfrak g}\) is rigid. In the paper under review cohomologies of semi-direct products \({\mathfrak g}= \tau+{\mathfrak n}\), where \({\mathfrak n}\) is a nilpotent Lie algebra and \(\tau\) is a torus of derivations of \({\mathfrak n}\) are considered. \textit{R. Carles} [Ann. Inst. Fourier 34, No. 3, 65-82 (1984; Zbl 0544.17004)] proved that any solvable rigid Lie algebra will have this form. For a Lie algebra \({\mathfrak n}\) defined by generators and defining relations a formula for calculating invariants over the maximal torus \(H({\mathfrak n},{\mathfrak n})^ T\) is proposed. A Lie algebra \({\mathfrak g}\) is called pseudo-nilpotent, if the supplement of \([{\mathfrak g},{\mathfrak g}]\) in \({\mathfrak g}\) generates \({\mathfrak g}\) and \(\bigcap_{p\geq 1} C^ p{\mathfrak g}=0\), where \(C^ p{\mathfrak g}\) is the descending central series of \({\mathfrak g}\). A pseudo-nilpotent Lie algebra has maximal rank if the minimal number of generators is equal to the dimension \(s\) of the maximal torus. Such a Lie algebra \({\mathfrak m}\) is isomorphic to the quotient of a free Lie algebra \({\mathfrak L}_ s\) with \(s\) generators over an ideal contained in \([{\mathfrak L}_ s,{\mathfrak L}_ s]\) and stable under the action of the maximal torus \(T_ s\). If \(P\) is the set of weights of \(T_ s\) over \({\mathfrak m}\), then \({\mathfrak m}\) is the quotient by a \(T_ S\)-stable ideal of a Lie algebra admitting \(P\) as a set of weights and is maximal according to this property. This Lie algebra corresponding to \(P\) is called a Lie model associated to \(P\). For any model \({\mathfrak m}\) the group \(H^ k({\mathfrak m},{\mathfrak m})\) is zero for \(k=0,2\) and equal to \(T\) for \(k=1\). In particular, \(H^ k({\mathfrak g},{\mathfrak g})=0\) for \(k\leq 2\), if \(…
reviewed by: A. Dzhumadil'daev
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 168584
zbMATH Keywords: cohomology of Lie algebras
zbMATH Keywords: nilpotent algebras
zbMATH Keywords: Borel subalgebras
zbMATH Keywords: deformations
zbMATH Keywords: rigid algebras
zbMATH Keywords: cohomologies of semi-direct products
zbMATH Keywords: pseudo-nilpotent Lie algebra
zbMATH Keywords: Lie model
Wikidata QID: Q115396331
MaRDI profile type: Publication
full work available at URL: jabr.1993.1016
OpenAlex ID: W2041012540
DOI: 10.1006/JABR.1993.1016
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description: scientific article; zbMATH DE number 168584