# Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $$\Bbb C$$-linear involutions of simple complex Lie algebras?

Question 2. If not, what kind of a similar/related classification does it contain?

Question 3. In what book/paper is this Cartan's paper discussed?

MathSciNet contains only the title, while zbMATH contains a review: a translation into German of a few lines from the introduction.

• I do read French. However, it is hard to read a paper on Lie groups or Lie algebras written before Dynkin: the language has changed completely! Jun 30, 2020 at 12:24
• Why are you asking about $\mathbf{C}$-linear involutions? doesn't the topic of the paper suggest that it classifies $\mathbf{C}$-skew-linear involution? In any case it is never explicit (the paper doesn't mention automorphisms). And all along it makes this classification, which is equivalent to classifying those skew-linear involutions (more than half of the paper is about exceptional cases EFG). I don't think it gives any information on linear involutions.
– YCor
Jun 30, 2020 at 14:20
• About Q3: scholar.google.fr/… can always help.
– YCor
Jun 30, 2020 at 14:22
• @YCor: You are right. We need real structures (anti-linear involutions). However, conjugacy classes of of real structures bijectively correspond to conjugacy classes of $\Bbb C$-linear involutions. See page 442 of Helgason's book, or Serre, Cohomologie galoisienne, III.4.5, Theorem 6 and Example (b). Helgason writes that Cartan discovered this later, in 1929. Jun 30, 2020 at 17:10

## 2 Answers

The paper and its progeny are discussed at length in Helgason (1978, p. 537):

In his paper [2] Cartan classifies the simple Lie algebras over R. His method, which required formidable computations, used the signature of the Killing form although it often happens that two nonisomorphic real forms of the same complex algebra have the same signature. Cartan's statement ([2], p. 263): “Les groupes réels d'ordre $$r$$ qui correspondent à une même type complexe d’ordre $$r$$ se classent en général complètement d’après leur caractère,” is therefore not to be taken literally; cf. Lardy ([1], p. 195). After noticing the equivalence of problems B and B' (§1) Cartan (in [12]) simplified his original treatment (see also Lardy [1]). Following his general theory [1] of automorphism of complex simple Lie groups, Gantmacher [2] gave a simplified treatment of the real classification. For further developments of this method see Murakami [3], Wallach [2], and Freudenthal and de Vries [1]. While Gantmacher used a Cartan subalgebra $$\mathfrak h$$ of $$\mathfrak g$$ whose “toral part” $$\mathfrak h\cap\mathfrak k$$ is maximal abelian in $$\mathfrak k$$, Araki develops in [1] a new method using a Cartan subalgebra $$\mathfrak h\subset\mathfrak g$$ whose “vector part” $$\mathfrak h\cap\mathfrak p$$ is maximal abelian in $$\mathfrak p$$. In addition to a solution to problem B' (§1) this method gives valuable information about the restricted roots and their multiplicities (cf. Exercises F). A modification is given by Sugiura [2]. In the present work we use the method of Kac [1] which at the same time gives a rather explicit description of the automorphism $$\sigma$$ of finite order.

• Excellent! Thank you! I have Helgason's book in my bookcase, but I have not read it.... Jun 30, 2020 at 16:52
• Just a comment about Helgason's statement "Cartan's statement...is therefore not to be taken literally". I always felt that this was a little unfair. When Cartan wrote "en général" in that sentence, I feel that he intended it to be read as 'usually' or as we would use the phrase 'it is generally the case that'. Cartan obviously knew examples of non-isomorphic real forms with the same character, but usually the character is enough to distinguish two real forms of the same complex simple Lie algebra. In fact, I think he was alerting the reader to the fact that it was not always sufficient. Jun 30, 2020 at 18:45
• @RobertBryant Fair statement (on the statement (on the statement))! ;) Jul 2, 2020 at 3:00

The paper of Cartan is discussed in

D. W. Morris, Introduction to arithmetic groups. Deductive Press, [place of publication not identified], 2015.

The free book can be downloaded from arXiv: https://arxiv.org/abs/math/0106063.

Here is a quote from that book:

The classification of real simple Lie algebras (Theorem 18.1.7) was obtained by E. Cartan [4]. (The intervening decades have led to enormous simplifications in the proof.)

• The book discusses the involutive isometries of symmetric spaces, and Cartan involutions due to Cartan subgroups, but doesn't say whether Cartan ever mentioned them. Jun 30, 2020 at 14:24