Triality, exceptional Lie algebras and Deligne dimension formulas.
type: Item
instance of: scholarly article
title: Triality, exceptional Lie algebras and Deligne dimension formulas.
zbMATH Open document ID: 1035.17016
author: Laurent Manivel
author: J. M. Landsberg
published in: Advances in Mathematics
publication date: 2003-03-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0107032
review text: \textit{P. Deligne} [C. R. Acad. Sci., Paris, Sér. I 322, 321--326 (1996; Zbl 0910.22008)] observed some uniform representation-theoretic behaviour shared by exceptional Lie groups and found a small family of representations which behave in the same way. In particular, the dimensions of these representations are rational functions involving only linear factors. Using a computer program, \textit{A. M. Cohen} and \textit{R. de Man} [C. R. Acad. Sci., Paris, Sér. I 322, 427--432 (1996; Zbl 0849.22017)] found some computational evidence and extended the Deligne formulas to \({\mathfrak g}^{\otimes 3}\) and \({\mathfrak g}^{\otimes 4}\) where \({\mathfrak g}\) is the corresponding Lie algebra and \textit{P. Deligne} and \textit{R. de Man} [C. R. Acad. Sci., Paris, Sér. I 323, 577--582 (1996; Zbl 0910.22009)] refined the framework for these representations. In the paper under review the authors extend the above results in a computer free way. They give additional dimension formulas for the exceptional series as well as beautiful uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. The authors use a version of the Vinberg construction of the exceptional Lie algebras in terms of normed division algebras. They also use the triality model of the magic square of Freudenthal, which they review, and present a simplified proof of its validity. The paper concludes with discussions on how to obtain other series of Lie algebras equipped with series of representations of dimensions which are rational functions involving linear factors only.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B25
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 1888317
zbMATH Keywords: exceptional Lie algebras
zbMATH Keywords: Deligne dimension formulas
zbMATH Keywords: Freudenthal magic square
zbMATH Keywords: triality
describes a project that uses: LiE
MaRDI profile type: Publication
OpenAlex ID: W1974297348
arXiv ID: math/0107032
cites work: A class of nonassociative algebras with involution containing the class of Jordan algebras
cites work: Q5580319
cites work: Q4879949
cites work: Q4878340
cites work: Q4715857
cites work: Lie groups in the foundations of geometry
cites work: Q4040877
cites work: The projective geometry of Freudenthal's magic square.
cites work: Construction and classification of complex simple Lie algebras via projective geometry
cites work: On the ideals of secant varieties of Segre varieties
DOI: 10.1006/AIMA.2002.2071
description: scientific article