Gradings on simple Jordan and Lie algebras
type: Item
instance of: scholarly article
title: Gradings on simple Jordan and Lie algebras
zbMATH Open document ID: 1066.17018
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: The simple classical Lie algebras over an algebraically closed field \(F\) of characteristic \(0\) consist of either the zero trace \(n\times n\) matrices for \(n\geq 2\), or of the skew symmetric elements in \(M_n(F)\) relative to an orthogonal or symplectic involution. \textit{Yu. A.~Bahturin} and \textit{M. V.~Zaicev} [Can. Math. Bull. 45, 499--508 (2002; Zbl 1022.16032)] determined the gradings on \(M_n(F)\). Their results are instrumental here. In the paper under review, the gradings on matrix algebras with involutions (that is, gradings where the homogeneous components are invariant under the involution) are determined first. Then this is used to determine all the gradings by finite abelian groups of the simple classical Lie algebras other than \(D_4\), and of the simple special Jordan algebras other than the Jordan algebras of symmetric nondegenerate bilinear forms. Gradings by torsion free groups are studied too.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 17C50
Mathematics Subject Classification ID: 17C70
zbMATH DE Number: 2136612
zbMATH Keywords: Gradings
zbMATH Keywords: involution
zbMATH Keywords: simple classical Lie algebra
zbMATH Keywords: simple special Jordan algebra.
Wikidata QID: Q115351690
author: Yuri Bahturin
author: Ivan Shestakov
author: Mikhail Zaicev
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.10.007
OpenAlex ID: W2035841452
cites work: Group Gradings on Matrix Algebras
cites work: Group gradings on associative algebras
cites work: Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices
cites work: Q5569204
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q5622399
cites work: Graded Lie algebras and symmetric spaces
cites work: Q3998356
cites work: The eight fine gradings of sl(4, C) and o(6, C)
DOI: 10.1016/J.JALGEBRA.2004.10.007
Recommended article: Gradings on simple Lie algebras
Recommended article: Jordan gradings on exceptional simple Lie algebras
Recommended article: Gradings on finite-dimensional simple Lie algebras
Recommended article: Q5457595
Recommended article: GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS
Recommended article: Gradings of positive rank on simple Lie algebras
Recommended article: Q4672428
Recommended article: Fine gradings on simple classical Lie algebras
Recommended article: Non-group gradings on simple Lie algebras
Recommended article: On the fine gradings of simple classical Lie algebras
description: scientific article; zbMATH DE number 2136612