Root strings with three or four real roots in Kac-Moody root systems
type: Item
instance of: scholarly article
title: Root strings with three or four real roots in Kac-Moody root systems
zbMATH Open document ID: 0656.17012
author: Jun Morita
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: The author gives a necessary and sufficient condition, in terms of generalized Cartan matrices, for a Kac-Moody root system to have a root string with three or four real roots. Let \(A=(a_{ij})\) be an \(n\times n\) generalized Cartan matrix, \(\Delta\) the associated root system, and \(\Delta^{re}\) the set of real roots. Put \(r(\alpha;\beta)=\#| \{\beta +k\alpha | \quad k\in {\mathbb{Z}}\}\cap \Delta^{re}|\) for \((\alpha,\beta)\in \Delta^{re}\times \Delta.\) Then the following two conditions are equivalent. (1) \(r(\alpha;\beta)=3\) or 4 for some \((\alpha,\beta)\in \Delta^{re}\times \Delta\). \((2)\quad a_{ij}=- 1\quad and\quad a_{ji}<-1\) for some i,j (1\(\leq i,j\leq n).\) As a corollary, one can simplify the Steinberg-Tits presentation of the associated Kac-Moody group in the case when A has a certain property.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 4071148
zbMATH Keywords: root system
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: generalized Cartan matrices
zbMATH Keywords: Kac-Moody root system
zbMATH Keywords: root string
zbMATH Keywords: Steinberg-Tits presentation
zbMATH Keywords: Kac-Moody group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227928
OpenAlex ID: W2057458818
cites work: Infinite dimensional Lie algebras. An introduction
cites work: Q3797354
cites work: Root Systems and Cartan Matrices
cites work: Commutator relations in Kac-Moody groups
cites work: Q4064209
cites work: Uniqueness and presentation of Kac-Moody groups over fields
DOI: 10.2748/TMJ/1178227928
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description: scientific article; zbMATH DE number 4071148