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Abstract simplicity of locally compact Kac–Moody groups

Part of:Structure and classification of infinite or finite groupsLinear algebraic groups and related topics

Published online by Cambridge University Press:  10 March 2014

Timothée Marquis
Timothée Marquis*
Affiliation:
UCL, 1348 Louvain-la-Neuve, Belgium email timothee.marquis@uclouvain.be
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Abstract

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In this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.

Keywords

abstract simplicitycomplete Kac–Moody groupstopological Kac–Moody groupscontraction groupsroot groups

MSC classification

Secondary:20E32: Simple groups20E42: Groups with a $BN$-pair; buildings20G44: Kac-Moody groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014, pp. 713 - 728
Copyright
© The Author 2014 

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