On a Theorem of E. Cartan

Ichiro SATAKE

doi:10.2969/jmsj/00230284

Ichiro SATAKE

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1951 Volume 2 Issue 3-4 Pages 284-305

DOI https://doi.org/10.2969/jmsj/00230284

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Published: 1951 Received: October 30, 1950 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) The numbers in brackets refer to the bibliography at the end of the paper.
2) See Weyl [10], K. IV, §§ 3, 4.
3) See Weyl [11], Ch. III, §§ 4, 5.
4) See Weyl [10], K. IV, §§ 1, 2. Recently H. Chandra [4] gave a new proof of Weyl′ stheorem as a consequence of Cartan′s theorem.
5) Cf. van der Waerden [9] . He added to the definition of root systems, a supplementary condition that under the same assumption as our Lemma 1, all the following vectors β, β-εα, β-2εα,…,β-2(α,β)/(α,α) alpha; (=S_αβ) also belong to r (cf. (1.5)). This condition, however, is unnecessary. For we can prove it bythe successive applications of Lemma 1 starting from β and S_αβ.
6) sign(α, λ) is+1,0,-1 according as (α, λ)>,=,<0.
7) The proof is analytical in spite of the algebraic nature of Proposition 1 and a more direct (and algebraical !) proof seems to be desirable. Cf. Remark at the end of § 6.
8) Cf. Cartan [1], or van der Waerden [9]. See also § 3 of the present paper.
9) Conversely, if two irreducible root systems have one and the same transformation group in the above sense, they are congruent or inverse to each other. From this point of view, H. Weyl ([11], Ch. III, § 6) gave a new classification-theory.
10) δ_ij is Kronecker′s delta.
11) The symbol-denotes the set-theoretical difference.
12) Weyl proved Prop. 2 as a consequence of this Corollary ([11], Ch. III, §§ 4, 5).
13) The symbol φ denotes the empty set.
14) 1 denotes the identical transformation of Eⁿ.
15) A direct proof is given in Weyl [11], Ch. III, § 4.
16) Cf. Gantmacher [7], Ch. III.
17) Our schemata are quite analogous to those of Coxeter ([6], or Weyl [11], Ch. III, § 6).
18) It seems to me that the algebraic source of the analogies between the groups of type B_n and C_n, such as the coincidence of the Poincaré groups of their adjoint groups or hat of the Poincaré polynomials of their compact forms, lies in this very point (see Cor, to Prop. 6 in § 6).
19) SL(n+1, C) is the Lie algebra of all complex matrices of degree n+1 with vanishing traces.
20) As is seen from (4.7), the 1-st and n-th fundamental representations are realized by SL(n+1, C), where the matrices representing the same element of A_n are contragredient to each other. (This is another proof of the fact that the outer automorphism group of A_n consists of two elements.) The Cartan composite of these two representations is the adjoint representation. The i-th and (u+1-i)-th fundamental representations are the irreducible representations of SL(n+1, C) indicated by the so-called Young′s diagram _??_……These results are simple applications of Cartan′s Theorem I.
21) For fundamental concepts on Lie groups, see, for example, Chevalley [5].
22) As L is semi-simple, the adjoint group _??_₀ is one of the Lie groups corresponding to L, Remark also that _??_₀ is the adjoint group of L.
23) This will be proved afterwards to he always true.
24) See Chevalley [5], Ch. VI.
25) That _??_⊆_??_₁ can be proved directly. Cf. Weyl [11], Ch. III, § 5, or Gantmacher [7], Ch. III.
26) Concerning the notion of covering space we also refer to Chevalley [5], Ch. II.
27) Cf. Weyl [11], Ch. III, § 5.
28) Ξ can be proved to be a simplex, when L is a simple Lie algebra. Cf. Cartan [3], or Weyl [11], Ch. III, § 5.
29) See Weyl [10], K. IV, § 1.
30) u₁ denotes the left coset of u₁ in the homogeneous space of _??_ modulo _??_ (or Z(_??_)). When u∈V(_??_), u₁u^-1 is uniquely determined by u₁.

Right : [1] Cartan, E.: Sur la structure des groupes de transformations finis et continus, Paris Dissertation, 1894.
[2] Cartan, E.: Les groupes projectifs qui ne laissent invariante aucune multiplicite plane, Bull. Soc. Math., t. 41 (1913).
[3] Cartan, E.: La géométrie des groupes simples, Annali di Mat., t. 4 (1927).
[4] Chandra, H.: Lie algebras and the Tannaka duality theorem, Ann. of Math., v. 51 (1950).
[5] Chevalley, C.: The theory of Lie groups, I, Princeton, 1946.
[6 ] Coxeter, H. S. M.: Discrete groups generated by reflections, Ann. of Math., v. 35 (1934).
[7] Gantmacher, F.: Canonical representation of automorphisms of a complex semi-simple Lie algebra, Rec. Math., t. 5 (1939).
[8] Peter, F. and Weyl, H.: Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann., v. 97 (1927).
[9] Waerden, B. L. van der: Die Klassifikation der einfachen Lieschen Gruppen, Math. Zeitschr., v. 37 (1933).
[10] Weyl, H.: Theorie der Darstellung kontinuierlicher halb-einfachér Gruppen durch lineare Transformationen, I, II, III, Math. Zeitschr., v. 23-24 (1925-26).
[11] Weyl, H.: The structure and representations of continuous groups, Lectures at the Institute for Advanced Study, Princeton, 1934-35.
[12] Weyl, H.: The classical groups, Princeton, 1939.

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