Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T02:33:49.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Abstract simplicity of locally compact Kac–Moody groups

Part of: Structure and classification of infinite or finite groups Linear 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 [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 groups 20E42: Groups with a $BN$-pair; buildings 20G44: Kac-Moody groups
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 713 - 728
Copyright
© The Author 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramenko, P. and Brown, K. S., Buildings, in Theory and applications, Graduate Texts in Mathematics, vol. 248 (Springer, New York, 2008).Google Scholar
Baumgartner, U., Ramagge, J. and Rémy, B., Contraction groups in complete Kac-Moody groups, Groups Geom. Dyn. 2 (2008), 337–352.Google Scholar
Caprace, P.-E., Abstract homomorphisms of split Kac–Moody groups, Mem. Amer. Math. Soc. 198 (2009), xvi+84.Google Scholar
Carbone, L., Ershov, M. and Ritter, G., Abstract simplicity of complete Kac-Moody groups over finite fields, J. Pure Appl. Algebra 212 (2008), 2147–2162.Google Scholar
Carbone, L. and Garland, H., Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003), 813–867.Google Scholar
Caprace, P.-E. and Monod, N., Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), 97–128.CrossRefGoogle Scholar
Caprace, P.-E. and Rémy, B., Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009), 169–221.Google Scholar
Caprace, P.-E. and Rémy, B., Simplicity of twin tree lattices with non-trivial commutation relations, Preprint (2012), arXiv:1209.5372 [math.GR].Google Scholar
Capdeboscq, I. and Rémy, B., On some pro-$p$groups from infinite-dimensional Lie theory, Preprint (2013), arXiv:1302.4174 [math.GR].Google Scholar
Caprace, P.-E., Reid, C. and Willis, G., Limits of contraction groups and the Tits core, Preprint (2013), arXiv:1304.6246 [math.GR].Google Scholar
Kac, V. G., Infinite-dimensional Lie algebras, third edition (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
Kumar, S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204 (Birkhäuser, Boston, 2002).Google Scholar
Mathieu, O., Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 227–230.Google Scholar
Moody, R., A simplicity theorem for Chevalley groups defined by generalized Cartan matrices, Preprint (April 1982).Google Scholar
Rémy, B., Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups, Geom. Funct. Anal. 14 (2004), 810–852; with an appendix by P. Bonvin.Google Scholar
Rousseau, G., Groupes de Kac–Moody déployés sur un corps local, II masures ordonnées, Preprint (2012), arXiv:1009.0138 [math.GR].Google Scholar
Rémy, B. and Ronan, M., Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv. 81 (2006), 191–219.CrossRefGoogle Scholar
Speyer, D. E., Powers of Coxeter elements in infinite groups are reduced, Proc. Amer. Math. Soc. 137 (2009), 1295–1302.Google Scholar
Springer, T. A., Linear algebraic groups, second edn, Progress in Mathematics, vol. 9 (Birkhäuser, Boston, MA, 1998).Google Scholar
Tits, J., Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra 105 (1987), 542–573.Google Scholar
Tits, J., Groupes associés aux algèbres de Kac-Moody, in Astérisque; Séminaire Bourbaki, Vol. 1988/89 (1989), 7–31; no. 177–178, Exp. No. 700.Google Scholar
Wang, J. S. P., The Mautner phenomenon for p-adic Lie groups, Math. Z. 185 (1984), 403–412.CrossRefGoogle Scholar
Willis, G., The nub of an automorphism of a totally disconnected, locally compact group, Ergodic Theory Dynam. Systems, to appear (2012).Google Scholar