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On algebraic groups defined by Jordan pairs

Published online by Cambridge University Press:  22 January 2016

Ottmar Loos
Ottmar Loos*
Affiliation:
Department of Mathematics, University of British Columbia
*
Current address: Institut für Mathematik Universität InnsbruckInnsbruck, Austria
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Let G be an algebraic group over a field k, and let ψ be an action of the multiplicative group km of k on G by automorphisms. We say ψ is an elementary action if it has only the weights 0, ±1; more precisely, if there exist subgroups H, U+, U- of G such that (i) H is fixed under ψ, (ii) U+ and U+ are vector groups and (iii) Ω = U+. H . U+ is open in G, and (iv) G is generated by H, U+, U+. This situation is characteristic for the complexifications of the automorphism groups of bounded symmetric domains (see, e.g., [9, 16]). A typical example is G = GLnwith (matrices being decomposed into 4 blocks) ψ given by

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 74 , July 1979 , pp. 23 - 66
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Anantharaman, S., Schémas en groupes, espaces homogènes, et espaces algébriques sur une base de dimension 1. Bull. Soc. Math. France, Meroire 33 (1973), 5–79.Google Scholar
[2] Borel, A., Linear algebraic groups. W.A. Benjamin, New York-Amsterdam 1969.Google Scholar
[3] Borel, A. and Tits, J., Groupes réductifs. Publ. Math. IHES No. 27 (1965), 55–150.Google Scholar
[4] Borel, A. and Tits, J., Homomorphismes “abstraits” de groupes algébriques simples. Ann. of Math. 97 (1973), 499–571.Google Scholar
[5] Demazure, M. and Gabriel, P., Groupes algébriques. Masson, Paris 1970.Google Scholar
[6] Faulkner, J., Octonion planes defined by quadratic Jordan algebras. Memoirs Amer. Math. Soc. No. 104, 1970.Google Scholar
[7] Grothendieck, A. and Demazure, M., Schémas en groupes. Springer Lecture notes 151–153, 1970. (Referred to as SGA3)Google Scholar
[8] Grothendieck, A. and Dieudonné, J., Eléments de géométrie algébrique. Publ. Math. IHES 1960-…. (Referred to as EGA)Google Scholar
[9] Helgason, S., Differential Geometry and Symmetric Spaces. Academic Press, 1962.Google Scholar
[10] Koecher, M., Imbedding of Jordan algebras into Lie algebras I, II. Amer. J. Math. 89 (1967), 787–816, and 90 (1968), 476–510.Google Scholar
[11] Koecher, M., Über eine Gruppe von rationalen Abbildungen. Inv. Math. 3 (1967), 136–171.Google Scholar
[12] Koecher, M., Gruppen und Lie-Algebren von rationalen Funktionen. Math. Z. 109 (1969), 349–392.Google Scholar
[13] Koecher, M., An elementary approach to bounded symmetric domains. Lecture notes, Rice University, Houston 1969.Google Scholar
[14] Loos, O., Representations of Jordan triples. Trans. Amer. Math. Soc. 185 (1973), 199–211.Google Scholar
[15] Loos, O., Jordan pairs. Springer Lecture notes No. 460, 1975.CrossRefGoogle Scholar
[16] Loos, O., Jordan pairs and bounded symmetric domains. Lecture notes, University of California, Irvine 1977.Google Scholar
[17] Meyberg, K., Jordan-Tripelsysteme und die Koecher-Konstruktion von Lie-Algebren. Math. Z. 115 (1970), 58–78.Google Scholar
[18] Petersson, H. P., Zur Arithmetik der Jordan-Paare. Math. Z. 147 (1976), 139–161.Google Scholar
[19] Roby, N., Lois polynomes et lois formelles en theorie des modules. Ann. Scient. Ecole Norm. Sup. 3e serie, t. 80 (1963), 213–348.Google Scholar
[20] Springer, T. A., Jordan algebras and algebraic groups. Springer Verlag 1973.Google Scholar
[21] Tits, J., Une classe d’algèbres de Lie en relation avec les algebres de Jordan. Proc. Kon. Akad. Wet. Amst. 65 (1962), 530–535.Google Scholar