A non-computational approach to the gradings on \(\mathfrak f_4\)
type: Item
instance of: scholarly article
title: A non-computational approach to the gradings on \(\mathfrak f_4\)
zbMATH Open document ID: 1257.17039
DOI: 10.4171/RMI/677
author: Cristina Draper Fontanals
published in: Revista Matemática Iberoamericana
publication date: 2012-03-03T00:00:00Z
review text: A grading by an abelian group \(G\) of a Lie algebra \({\mathcal L}\) is a decomposition \(\Gamma: {\mathcal L}=\oplus_{g\in G}{\mathcal L}_g\) as a direct sum of vector subspaces, such that \([{\mathcal L}_g,{\mathcal L}_h]\subseteq {\mathcal L}_{gh}\) for any \(g,h\in G\). Given two gradings \(\Gamma: {\mathcal L}=\bigoplus_{g\in G}{\mathcal L}_g\) and \(\Gamma': {\mathcal L}=\oplus_{h\in H}{\mathcal L}_h'\), \(\Gamma\) is a refinement of \(\Gamma'\) (or \(\Gamma'\) a coarsening of \(\Gamma\)) if any homogeneous space of \(\Gamma'\) is a sum of homogeneous spaces in \(\Gamma\). \(\Gamma\) is fine if it admits no proper refinement. Any grading is a coarsening of a fine grading. Over algebraically closed fields of characteristic zero, the classification of the fine gradings for the classical Lie algebras was completed by the reviewer [J. Algebra 324, No. 12, 3532--3571 (2010; Zbl 1213.17030)], for \(G_2\) it was obtained independently by \textit{C.~Draper} and \textit{C.~Martín} [Linear Algebra Appl. 418, No. 1, 85--111 (2006; Zbl 1146.17027)] and by \textit{Y.~A.~Bahturin} and \textit{M.~V.~Tvalavadze} [Commun. Algebra 37, No. 3, 885--893 (2009; Zbl 1190.17005)]. For \(F_4\) it was obtained by \textit{C.~Draper} and \textit{C.~Martín} [Rev. Mat. Iberoam. 25, No. 3, 841--908 (2009; Zbl 1281.17035)] in a very technical paper based on many concrete computations. The paper under review gives a different proof of the classification of the fine gradings on \(F_4\). The homogeneous components of any such grading are given by the common eigenspaces for the action of the elements of a maximal quasitorus of the algebraic group \(\text{Aut}({\mathcal L})\). Known results about algebraic groups are used to check that any such quasitorus is either a maximal torus or contains either a nontoral \(2\)-group or a nontoral \(3\)-group. In case it contains a nontoral \(2\)-group, the quasitorus is shown to live in a subgroup of \(\text{Aut}({\mathcal L})\) isomorphic to \(\text{Spin}…
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B25
zbMATH DE Number: 6011516
zbMATH Keywords: fine grading
zbMATH Keywords: exceptional Lie algebra
zbMATH Keywords: \(F_4\)
zbMATH Keywords: maximal abelian diagonalizable subgroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6011516