Graded Lie algebras with classical reductive null component
type: Item
instance of: scholarly article
title: Graded Lie algebras with classical reductive null component
zbMATH Open document ID: 0648.17005
DOI: 10.1007/BF01442673
author: Thomas Gregory
author: Georgia Benkart
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164586
review text: In this work we investigate finite-dimensional graded Lie algebras \(G=\oplus _ iG_ i\) over an algebraically closed field of characteristic \(p>5\) whose null component \(G_ 0\) is classical reductive (the direct sum of ideals each of which is abelian, classical simple, or sl(n), gl(n), \(gl(n)/F_ 1\) with p \(| n)\). In 1970 \textit{V. G. Kac} [Izv. Akad. Nauk SSSR, Ser. Mat. 34, 385-408 (1970; Zbl 0254.17007)] proved that if such an algebra G satisfies 4 additional constraints, then it is a classical Lie algebra with a standard grading, \(gl(m)/F_ 1\) where p \(| m\), or a graded Lie algebra of Cartan type. Kac's theorem, the so-called ``Recognition Theorem'', is an important ingredient in the recent classification of the restricted simple Lie algebras by Block and Wilson. The main result of our paper shows that one of Kac's assumptions, the representation of \(G_ 0\) on \(G_{-1}\) is restricted, is unnecessary. It is expected that this new result will have applicability in the classification of the nonrestricted finite-dimensional simple Lie algebras over algebraically closed fields of prime characteristic.
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 4057738
zbMATH Keywords: finite-dimensional graded Lie algebras
zbMATH Keywords: Recognition Theorem
zbMATH Keywords: nonrestricted finite-dimensional simple Lie algebras over algebraically closed fields
zbMATH Keywords: prime characteristic
MaRDI profile type: Publication
cites work: Determination of the differentiably simple rings with a minimal ideal
cites work: Classification of restricted simple Lie algebras
cites work: Simple Lie algebras with classical reductive null component
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q5732712
cites work: Q5583129
cites work: Q4039778
cites work: On the structure of the minimal ideal of some graded Lie algebras in characteristic \(p>0\)
cites work: On Subalgebras of Simple lie Algebras of Characteristic p > 0
cites work: A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic
description: scientific article; zbMATH DE number 4057738