K1 of Chevalley groups are nilpotent

Roozbeh Hazrat, N. Vavilov

Research output: Contribution to journalArticle

59 Citations (Scopus)

Abstract

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.
Original languageEnglish
Pages (from-to)99-116
Number of pages18
JournalJournal of Pure and Applied Algebra
Volume179(1-2)
Issue number1-2
DOIs
Publication statusPublished - 01 Apr 2003

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Chevalley Groups
Quotient
Root System
Commutative Ring
Simplification
Completion
Subgroup
Series

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Hazrat, Roozbeh ; Vavilov, N. / K1 of Chevalley groups are nilpotent. In: Journal of Pure and Applied Algebra. 2003 ; Vol. 179(1-2), No. 1-2. pp. 99-116.
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K1 of Chevalley groups are nilpotent. / Hazrat, Roozbeh; Vavilov, N.

In: Journal of Pure and Applied Algebra, Vol. 179(1-2), No. 1-2, 01.04.2003, p. 99-116.

Research output: Contribution to journalArticle

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