On the theory of reductive algebraic groups over a perfect field

Ichiro SATAKE

doi:10.2969/jmsj/01520210

Ichiro SATAKE

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1963 年 15 巻 2 号 p. 210-235

DOI https://doi.org/10.2969/jmsj/01520210

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Published: 1963 Received: 1962/09/20 Available on J-STAGE: 2006/09/26 Accepted: - Advance online publication: - Revised: -
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訂正日: 2006/09/26 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Right : 1) An algebraic group G, defined over k, is said to be a semi-direct product over k of closed subgroups G₁, G₂, both defined over k, if G is a semi-direct product of G₁, G₂ in the abstract sense and if the natural correspondence G₁×G₂→G is a birational (hence biregular) map defined over k.
2) Professor Tits has kindly informed me that this problem was already settled affirmatively by him.
3) According to the Tits's result (footnote 2)), the prefix ‘quasi-’ is superfluous.
4) Cf. R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math., 9 (1959), 875-891, D. Hertzig, Forms of algebraic groups, Proc. of Amer. Math. Soc., 12 (1961), 657-660, and S. Araki, On root systems and an infinitesimal classification of irreducible symmetric domains, J. of Math. Osaka City Univ., 13 (1963), 1-34. On the other hand, over a field of characteristic 0 and for classical groups (except D₄), Problem 2 can be solved by a result of A. Weil, (J. of Indian Math. Soc., 24 (1960), 589-623), and for exceptional groups of type G₂, F₄ by a result of H. Hijikata (J. Math. Soc. Japan, 15 (1963), 159-164).
5) The author was informed, after completion of this paper, that similar results as in §4 had been announced, without proof, by Tits [13], [14].
[1] A. Borel, Groupes linéaires algébriques, Ann. of Math., 64 (1956), 20-82.
[2] C. Chevalley, Théorie des groupes de Lie, III, Hermann, 1955.
[3] C. Chevalley, Sur certains groupes simples, Tohoku Math. J., 7 (1955), 14-66.
[4] C. Chevalley, Classification des groupes de Lie algébriques, t. 1-2, Séminaire C. Chevalley, 1956-58.
[5] R. Godement, Groupes linéaires algébriques sur un corps parfait, Séminaire Bourbaki, 1960-61, n° 206.
[6] T. Ono, Sur les groupes de Chevalley, J. Math. Soc. Japan, 10 (1958), 307-313.
[7] T. Ono, Arithmetic of algebraic tori, Ann. of Math., 74 (1961), 101-139.
[8] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math., 78 (1956), 401-443.
[9] M. Rosenlicht, Some rationality questions on algebraic groups, Ann. Math. Pura Appl., 43 (1957), 25-50.
[10] I. Satake, On algebraic groups over a p-adic field, (Japanese) Sugaku, 12 (1960), 195-202.
[11] A. Weil, On algebraic groups of transformations, Amer. J. Math., 77 (1955), 355-391.
[12] A. Weil, On algebraic groups and homogeneous spaces, Amer. J. Math., 493-512.
[13] J. Tits, Sur la classification des groupes algébriques semi-simples, C. R. Acad. Sci. Paris, 249 (1959), 1438-1440.
[14] J. Tits, Groupes algébriques semi-simples et géométries associées, Coll. on Algebraic and Topological Foundations of Geometry (Utrecht, 1959), Pergamon, 1962, 175-192.

訂正日: 2006/09/26 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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