Generators and relations for 3-graded Lie algebras
type: Item
instance of: scholarly article
title: Generators and relations for 3-graded Lie algebras
zbMATH Open document ID: 0769.17019
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: The main result of this paper can be viewed as being for Kantor-Koecher- Tits (KKT-) algebras of a Jordan pair spanned by a standard grid an analogue of the familiar construction of finite-dimensional semisimple Lie algebras using generators and relations based on root bases for root systems. (KKT-algebras are 3-graded Lie algebras \({\mathcal L}_ 1\oplus{\mathcal L}_ 0\oplus {\mathcal L}_{-1}\) for which \({\mathcal L}_ 0=[{\mathcal L}_ 1{\mathcal L}_{-1}]\) and \(\{x\in{\mathcal L}_ 0: [x{\mathcal L}_ 1]=[x{\mathcal L}_{-1}]=0\}=0\). (\(({\mathcal L}_ 1,{\mathcal L}_{-1})\) is a Jordan pair with respect to \(\{x^ \sigma y^{-\sigma} z^ \sigma\}=[[x^ \sigma y^{-\sigma}] z^ \sigma]\) for \(\sigma=\pm 1\) and \(x^ \sigma \in L_ \sigma\). A standard grid in a Jordan pair is a family of idempotents satisfying certain multiplication rules.) In this construction of KKT-algebras a root system \(R\) is allowed to lie in an arbitrary real inner product space, and root bases are replaced by grid bases \(B\), which are vector space bases contained in \(R\) such that every root in \(R\) is an alternating sum of roots in \(B\) satisfying another technical property. The special case when \(R\) is finite is stated as a Corollary. There is also an application to \(L^*\)-algebras.
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 168176
zbMATH Keywords: Kantor-Koecher-Tits algebras
zbMATH Keywords: Jordan pair
zbMATH Keywords: standard grid
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: 3-graded Lie algebras
zbMATH Keywords: idempotents
zbMATH Keywords: grid bases
author: Erhard Neher
MaRDI profile type: Publication
full work available at URL: jabr.1993.1029
OpenAlex ID: W1973727717
Wikidata QID: Q115396324
DOI: 10.1006/JABR.1993.1029
description: scientific article; zbMATH DE number 168176