Weyl groups of some fine gradings on \(\mathfrak{e}_6\)
type: Item
instance of: scholarly article
title: Weyl groups of some fine gradings on \(\mathfrak{e}_6\)
zbMATH Open document ID: 1325.17006
author: Diego Aranda Orna
author: Cristina Draper Fontanals
author: Valerio Guido
published in: Journal of Algebra
publication date: 2014-09-04T00:00:00Z
Mathematics Subject Classification ID: 17B25
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 6339761
zbMATH Keywords: fine grading
zbMATH Keywords: exceptional Lie algebra \(\mathfrak{e}_6\)
zbMATH Keywords: Weyl group
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.07.001
OpenAlex ID: W1992923615
cites work: Lectures on exceptional Lie groups. Ed. by Zafer Mahmud and Mamoru Mimura
cites work: The classification of \(p\)-compact groups for \(p\) odd
cites work: Gradings on the real forms of the Albert algebra, of g2, and of f4
cites work: Q3682663
cites work: Q3997915
cites work: Models of the Lie algebra \(F_4\)
cites work: Gradings on the Albert algebra and on \(\mathfrak{f}_4\)
cites work: The magic square and symmetric compositions
cites work: Gradings on symmetric composition algebras
cites work: Fine gradings and gradings by root systems on simple Lie algebras
cites work: Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra.
cites work: Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D
cites work: Gradings on simple Lie algebras
cites work: Q4002278
cites work: Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices
cites work: Q3892399
cites work: Q4040877
cites work: Gradings on finite-dimensional simple Lie algebras
cites work: Configurations of lines and models of Lie algebras
cites work: On Lie gradings. I
cites work: Fine grading of sl(p2,C) generated by tensor product of generalized Pauli matrices and its symmetries
cites work: Q5522244
cites work: Q4767466
cites work: Q5513844
cites work: THE WEYL GROUP OF A GRADED LIE ALGEBRA
cites work: Q3947798
DOI: 10.1016/J.JALGEBRA.2014.07.001
review text: The Weyl group of fine gradings with infinite universal grading group, on the exceptional simple Lie algebra \(E_6\) are computed. The ground field is supposed to be algebraically closed and of characteristic \(0\). The \(E_6\) Lie algebra has (up to equivalence) \(11\) fine gradings. There are \(7\) of these gradings whose universal grading group is finite and their study, by J. Yu, is almost complete. Thus the authors work in this paper with the complementary set of equivalence classes of fine gradings (ruling out the Cartan grading). The authors use a list of models suitable for their purposes. The list begins with the unified Tit's construction but includes some other models which, as far as this reviewer knows, are new. The constructions involve composition algebras, symmetric composition algebras and exceptional Jordan algebras. The six gradings under study are described in section 4. Roughly speaking they are: an inner \(\mathbb Z_2^3\times\mathbb Z^2\) grading denoted \(\Gamma_1\) and an outer \(\mathbb Z_2^3\times\mathbb Z^2\) grading denoted \(\Gamma_6\); a \(\mathbb Z_2^4\times\mathbb Z\) grading denoted \(\Gamma_2\); a \(\mathbb Z_2\times\mathbb Z^4\) grading denoted \(\Gamma_3\); a \(\mathbb Z_3^2\times\mathbb Z^2\) grading \(\Gamma_4\); and a \(\mathbb Z_2^5\times\mathbb Z\) grading \(\Gamma_5\). The Weyl groups are NEWLINE\[NEWLINEW(\Gamma_1)=\text{Mat}_{3\times 2}(\mathbb Z_2)\rtimes(GL_3(\mathbb Z_2)\times D_3)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_2)=\mathbb Z_2^4\rtimes(GL_3(\mathbb Z_2)\times \mathbb Z_2)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_3)=\mathbb Z_2^2\rtimes((\mathbb Z_2^3\rtimes S_4)\rtimes S_3)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_4)=\mathbb Z_3^2\rtimes(GL_2(\mathbb Z_3)\times D_6)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_5)=(SP_4(\mathbb Z_2)\times\mathbb Z_2)\ltimes \mathbb Z_2^5NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_6)=\mathbb Z_2^4\rtimes((\mathbb Z_2^2\rtimes S_3)\times (\mathbb Z_2^2\rtimes\mathbb Z_2)).NEWLINE\]
description: scientific article; zbMATH DE number 6339761