[Groupes de Steinberg pour les paires de Jordan]
Nous annonçons les résultats suivants relatifs aux groupes élémentaires projectifs et aux groupes de Steinberg associés aux paires de Jordan V munies d'une graduation par un système de racines Φ localement fini : Le groupe élémentaire projectif est un groupe avec des relations de commutateurs de type Φ par rapport à certains sous-groupes radiciels. Sous des conditions additionnelles faibles, le groupe de Steinberg associé à couvre de manière unique chaque extension centrale de et il est l'extension centrale universelle de si Φ est irréductible et de rang infini.
We announce results on projective elementary groups and on Steinberg groups associated to Jordan pairs V with a grading by a locally finite 3-graded root system Φ: The projective elementary group of V is a group with Φ-commutator relations with respect to appropriately defined root subgroups. Under some mild additional conditions, the Steinberg group associated to uniquely covers all central extensions of and is the universal central extension of if Φ is irreducible and has infinite rank.
Accepté le :
Publié le :
@article{CRMATH_2010__348_15-16_839_0,
author = {Loos, Ottmar and Neher, Erhard},
title = {Steinberg groups for {Jordan} pairs},
journal = {Comptes Rendus. Math\'ematique},
pages = {839--842},
publisher = {Elsevier},
volume = {348},
number = {15-16},
year = {2010},
doi = {10.1016/j.crma.2010.07.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.07.012/}
}
TY - JOUR AU - Loos, Ottmar AU - Neher, Erhard TI - Steinberg groups for Jordan pairs JO - Comptes Rendus. Mathématique PY - 2010 SP - 839 EP - 842 VL - 348 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.07.012/ DO - 10.1016/j.crma.2010.07.012 LA - en ID - CRMATH_2010__348_15-16_839_0 ER -
Loos, Ottmar; Neher, Erhard. Steinberg groups for Jordan pairs. Comptes Rendus. Mathématique, Tome 348 (2010) no. 15-16, pp. 839-842. doi : 10.1016/j.crma.2010.07.012. http://www.numdam.org/articles/10.1016/j.crma.2010.07.012/
[1] Groupes et algèbres de Lie, Masson, Paris, 1981 (Chapitres 4–6)
[2] Groups with Steinberg Relations and Coordinatization of Polygonal Geometries, Memoirs, vol. 185, Amer. Math. Soc., 1977
[3] Stable range and linear groups for alternative rings, Geom. Dedicata, Volume 14 (1983), pp. 177-188
[4] The Classical Groups and K-Theory, Grundlehren, vol. 291, Springer-Verlag, 1989
[5] An Elementary Approach to Bounded Symmetric Domains, Rice University, Houston, TX, 1969
[6] Jordan Pairs, Lecture Notes in Mathematics, vol. 460, Springer-Verlag, Berlin, 1975
[7] On algebraic groups defined by Jordan pairs, Nagoya Math. J., Volume 74 (1979), pp. 23-66
[8] Elementary groups and stability for Jordan pairs, K-Theory, Volume 9 (1994), pp. 77-116
[9] Steinberg groups and simplicity of elementary groups defined by Jordan pairs, J. Algebra, Volume 186 (1996) no. 1, pp. 207-234
[10] Locally finite root systems, Mem. Amer. Math. Soc., Volume 171 (2004) no. 811, p. x+214
[11] Systèmes de racines 3-gradués, C. R. Acad. Sci. Paris, Ser. I, Volume 310 (1990), pp. 687-690
[12] Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra, Volume 105 (1987), pp. 542-573
Cité par Sources :
