A Matsumoto-type theorem for Kac-Moody groups
type: Item
instance of: scholarly article
title: A Matsumoto-type theorem for Kac-Moody groups
zbMATH Open document ID: 0701.19001
author: Jun Morita
author: Ulf Rehmann
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1990-01-01T00:00:00Z
review text: A presentation of the \(K_ 2\) group associated with a Kac-Moody group over a field is given, which is a natural generalization of the result by \textit{H. Matsumoto} [Ann. Sci. Ecole Norm. Sup., IV. Ser. 2, 1-62 (1969; Zbl 0261.20025)] for a split semisimple algebraic group. Also a universal central extension of a Kac-Moody group over an infinite field is determined using the same method as \textit{R. Steinberg} [Lectures on Chevalley groups, Yale Univ. Lect. Note, New Haven, Conn. (1968)]. Hence the Schur multiplier of a (universal) Kac-Moody group over an infinite field is just the corresponding \(K_ 2\) group. The main result is as follows. Let \(A=(a_{ij})\) be an \(n\times n\) generalized Cartan matrix, F a field, and \(K_ 2(A,F)\) the \(K_ 2\) group associated with a Kac- Moody group over F defined by A. Then \(K_ 2(A,F)\) is isomorphic to the abelian group generated by the symbols \(c_ i(u,v)\) for \(1\leq i\leq n\) and u, v in the multiplicative group \(F^{\times}\) of F with the following defining relations: \(c_ i(u,v)c_ i(u,vw)=c_ i(uv,w)c_ i(v,w)\), \(c_ i(1,1)=1\), \(c_ i(u,v)=c_ i(u^{-1},v^{-1})\), \(c_ i(u,v)=c_ i(u,(1-u)v)\) if \(u\neq 1\), \(c_ i(u,v^{a_{ji}})=c_ j(u^{a_{ij}},v)\), \(c_{ij}(uv,w)=c_{ij}(u,w)c_{ij}(v,w)\) for all \(1\leq i\neq j\leq n\) and \(u,v,w\in F^{\times}\), where \(c_{ij}(u,vw)\) is defined to be \(c_ i(u,v)c_ j(u,w)\).
Mathematics Subject Classification ID: 19C20
Mathematics Subject Classification ID: 19C09
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 4148385
zbMATH Keywords: central group extensions
zbMATH Keywords: presentation of the \(K_ 2\) group associated with a Kac-Moody group over a field
zbMATH Keywords: universal central extension of a Kac-Moody group over an infinite field
zbMATH Keywords: Schur multiplier
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227573
OpenAlex ID: W2043567612
cites work: Q5580319
cites work: SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTH
cites work: Infinite dimensional Lie algebras. An introduction
cites work: Sur les sous-groupes arithmétiques des groupes semi-simples déployés
cites work: $ K$-COHOMOLOGY OF SEVERI-BRAUER VARIETIES AND THE NORM RESIDUE HOMOMORPHISM
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: A new class of Lie algebras
cites work: Tits' systems with crystallographic Weyl groups
cites work: Root Systems and Cartan Matrices
cites work: Commutator relations in Kac-Moody groups
cites work: Symplectic \(K_ 2\) of Laurent polynomials, associated Kac-Moody groups and Witt rings
cites work: Infinite flag varieties and conjugacy theorems
cites work: Q3341041
cites work: Q4064209
cites work: Uniqueness and presentation of Kac-Moody groups over fields
cites work: Q3485036
DOI: 10.2748/TMJ/1178227573
Recommended article: Q4797997
Recommended article: Q4505500
Recommended article: Kac-Moody Groups and Their Representations
Recommended article: Q5289095
Recommended article: On Koszul duality for Kac-Moody groups.
Recommended article: On the topology of Kac-Moody groups
Recommended article: Kac-Moody groups and completions
Recommended article: Q4696432
Recommended article: Kac-Moody groups and cluster algebras
description: scientific article; zbMATH DE number 4148385