Maximal subalgebras of simple modular Lie algebras
type: Item
instance of: scholarly article
title: Maximal subalgebras of simple modular Lie algebras
zbMATH Open document ID: 1106.17023
author: Hayk Melikyan
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: Let \(L\) be a restricted simple Lie algebra of Cartan type over an algebraically closed field of characteristic \(p>3\). The aim of the paper under review is to classify the maximal graded subalgebras \(M=M_{-2}\oplus M_{-1}\oplus M_0\oplus M_\oplus\cdots \oplus M_r\) of \(L\) where \(L\) has the standard grading \(L=L_{-2}\oplus L_{-1}\oplus L_0\oplus L_1\oplus\cdots\oplus L_r\) (and \(L_{-2}=0\) unless \(L\) is a contact algebra). A classification of those maximal graded subalgebras \(M\) of \(L\) (with \(L_{-2}=0\)) that contain \(L_{-1}\oplus L_0\) has previously been established by \textit{A. I. Kostrikin} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 251--322 (1969; Zbl 0211.05304)]. In the paper under review the author considers the remaining cases. He obtains an explicit construction unless \(M=L_{-2}\oplus L_{-1}\oplus M_0\oplus M_1\oplus\cdots\oplus M_r\) and \(M_0\) is a maximal subalgebra of \(L_0\) that acts irreducibly on \(L_{-1}\). In this case the problem is reduced to the classification of the irreducible maximal subalgebras of the simple classical Lie algebras which still is not complete.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 2153328
zbMATH Keywords: restricted simple Lie algebra of Cartan type
zbMATH Keywords: maximal graded subalgebra
zbMATH Keywords: simple classical Lie algebra
zbMATH Keywords: irreducible maximal subalgebra
Wikidata QID: Q115351683
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.11.001
OpenAlex ID: W2065760923
cites work: Q5580319
cites work: Q5842420
cites work: Semisimple subalgebras of semisimple Lie algebras
cites work: The maximal subgroups of the classical groups
cites work: Introduction to Lie Algebras and Representation Theory
cites work: ON THE CLASSIFICATION OF SIMPLE LIE ALGEBRAS OVER A FIELD OF NONZERO CHARACTERISTIC
cites work: DESCRIPTION OF FILTERED LIE ALGEBRAS WITH WHICH GRADED LIE ALGEBRAS OF CARTAN TYPE ARE ASSOCIATED
cites work: Q5807702
cites work: Q4775997
cites work: Q5540543
cites work: GRADED LIE ALGEBRAS OF FINITE CHARACTERISTIC
cites work: SIMPLE MODULAR LIE ALGEBRAS WITH A SOLVABLE MAXIMAL SUBALGEBRA
cites work: Witt algebra and linear groups
cites work: Q5840210
cites work: ON SIMPLE LIE ALGEBRAS OF CHARACTERISTIC 5
cites work: Q3234194
cites work: Q3310027
cites work: The maximal subgroups of classical algebraic groups
cites work: Maximal subgroups of exceptional algebraic groups
cites work: Simple Lie algebras over fields of positive characteristic. I: Structure theory
cites work: Q4039778
cites work: Q3758097
cites work: Q3765948
cites work: A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic
cites work: On Subalgebras of Simple lie Algebras of Characteristic p > 0
DOI: 10.1016/J.JALGEBRA.2004.11.001
Recommended article: Classification of restricted simple Lie algebras
Recommended article: The restricted simple Lie algebras are of classical or Cartan type
Recommended article: Maximal subalgebras of Lie superalgebras of Cartan type over fields of characteristic zero
Recommended article: Q3983558
Recommended article: Maximal subalgebras for modular graded Lie superalgebras of odd Cartan type
Recommended article: Maximal subalgebras for Lie superalgebras of Cartan type
Recommended article: Algebraic groups associated with graded Lie algebras of Cartan type
Recommended article: Q3826721
Recommended article: Graded Lie algebras with classical reductive null component
Recommended article: On the recognition theorem for Lie algebras of characteristic three
description: scientific article; zbMATH DE number 2153328