On algebraic Lie algebras

Morikuni Gotô

doi:10.2969/jmsj/00110029

Morikuni Gotô

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1948 年 1 巻 1 号 p. 29-45

DOI https://doi.org/10.2969/jmsj/00110029

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Published: 1948 Received: 1947/10/25 Available on J-STAGE: 2006/08/29 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/08/29 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) Maurer [1].
2) Chevalley and Tuan [1].
3) Chevalley [1].
4) Our l-algebraicity is the same as Chevalley and Tuan′s algebraicity.
5) Chevalley and Tuan [1]. Cf. also Matsushima [2].
6) Matsushima [2].
7) The condition of algebraic closedness is conventional, and all our theorems are valid for any field of characteristic 0.
8) Matsushima (in [1]) formulated this lemma in a somewhat different manner. The equivalence follows from the remark below.
9) Matsushima [1].
10) E. g. Zassenhaus [1].
11) Chevalley and Tuan [1].
12) Gantmacher [1].
13) Gantmacher (in [1]) gave an analogous theorem with respect to an element of a matric Lie group and the inner automorphiism defined by the element.
14) An inner derivation is called, regular if it has as many different eigenvalues as possible.
15) Chevalley and Tuan. 1. c.
16) For ordinary real (or complex) Lie algebras nilpotency is of course unnecessary.
17) Malcev. [1]. It seems to the writer that Malcev′s definition of inner automorphism is not very clear when K is a general field of characteristic 0.
18) Cartan [1].
19) Chevalley and Tuan 1. c.
20) Cartan [2].
21) Cartan [1].
22) Chevalley and Tuan [1]; cf also Goto [1].
23) This notion is due to Chevalley and Tuan.
24) Nilpotency is in fact unnecessary for our Lemma, and combining this to Lemma 7 we get a characterization of l-aig. Lie algebras. And analogous characterization of alg. Lie algebras (see § 5) may also be given easily. But the proof in the general case being rather cumbersome and superfluous for our present purpose, we limit ourselves to the nilpotent case.
25) Chevalley and Tuan [1].
26) See "Introduction"
27) Ado [1]; Cartan [3]. Recently K. Iwasawa gave a new proof on the basis of his theory of splitting Lie algebras. See his forthcoming paper in Jap. J. of Math.

Right : Ado, I. [1]: “On the representation of finite continuous groups by means of linear transformations” Izvestia Kazan, 7 (1934-35) (in Russian).
Cartan, E. [1]: Thèse, Paris (1894).
Cartan, E. [2]: “Les groupes de transformations continues, infinis, simples,” Ann, Ec. Norm. Sup., 3^e série, t. XXVI, (1909).
Cartan, E. [3]: “Les representations linéaires des groupes de Lie,” Jour. math. pures et app., 17 (1938).
Chevalley, C. [3]: “A new kind of relationship between matrices,” Amer. J. of Math., 65 (1943).
Chevalley, C. and Tuan, H. -F. [1] : “On algebraic Lie algebras,” Proc. Nat. Acad. Sci. U. S. A.
Gantmacher, F. [1]: “Canonical representation of automorphisms of a complex semi-simple Lie group,” Recuil Math. Moscou, 5 (1939).
Gotô, M. [1]: “On the replicas of nilpotent matrices,” forthcoming in Proc, Imp. Acad. Tokyo.
Malcev, A. [1]: “On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra,” Comptes Rendus (Doklady) (1942).
Matsushima, Y. [1]: “Note on the replicas of matrices,” forthcoming in Proc. Imp. Acad. Tokyo.
Matsushima, Y [2]: “On algebraic Lie groups and algebras,” forthcoming in this journal.
Maurer, L. [1]: “Zur Theorie der continuierlichen, homogenen und linearen Gruppen,” Sitz. d. Bayer. Acad., 24 (1894).
Zassenhaus, H. [1]: “Ueber Lie'sche Ringe mit Primzahlcharakteristik.” Hamb. Abh. 13 (1939).

訂正日: 2006/08/29 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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