On reflection groups and semisimple Lie algebras

Ulf Rehmann

Published in: Ernst Witt, Collected papers. Gesammelte Abhandlungen. Ed. by Ina Kersten. Springer-Verlag, Berlin, 1998, 247–255.

The classification of locally isomorphic semisimple Lie groups or, equivalently, of semisimple complex Lie algebras was obtained by the ingenious work of Killing [9] and of Cartan [2]. One of their main tools was the notion of the root system Φ of a semisimple complex Lie algebra 𝔤, as they called the collection of roots of the characteristic equation

(∗)  det(ad_{𝔤}(x) − T) = 0,

where x is a generic element of a Cartan subalgebra 𝔥 of 𝔤. The roots α ∈ Φ occur as linear forms on 𝔥 and, as such, generate a ℤ-lattice of full rank in the dual space 𝔥^{∗} of 𝔥.

While the original proofs by Killing and Cartan were quite complicated, a considerable simplification of the classification was achieved by a discovery of Weyl [8, II, p. 368, §4]. He observed the following. After 𝔥 is equipped with some natural euclidean metric (Killing form) and identified with its dual, some normal subgroup W of small index (= 1 in most cases) of the Galois group of (∗) is generated by all reflections s_{α} of 𝔥 at the hyperplane defined by α, were α runs over Φ. Then W stabilizes Φ. Clearly W is what we nowadays call the Weyl group of the root system Φ or of the semisimple Lie algebra.

A little later, v. d. Waerden, based on ideas of Schouten [10], simplified Weyl’s results and showed that the classification of semisimple complex Lie algebras is equivalent to the determination of reduced root systems [7, §2, p. 448], which are now given by an a priori description as subsets of a euclidean vector space in terms of their metric properties (cf. [21, ch. 6]). Of course, in v. d. Waerden’s proof the symmetries of the root system given by its Weyl group play an essential role.

On the other hand, groups generated by reflections (like W) had been studied and widely classified almost at the same time by Coxeter ([3] – [6], cf. also [24]).

Witt now, in his article [W], brings all the above notions together. He first gives an easy geometrical proof of Coxeter’s enumeration of all – finite and infinite – groups generated by euclidean reflections and describes them in terms of nowadays so called Coxeter diagrams [W, §3], [21, ch. 4, §1.9]. Witt calls a Coxeter diagram a “Figur”. The result is given in Satz 8, §4.

Ulf Rehmann

The geometrical idea of the proof is very intuitive and is described best by Satz 2. One of the remarkable features of the proof is the fact that finite and infinite reflection groups can be handled by the same method, and that both cases can be distinguished by the easy criterion given in Satz 6. The reflection group is finite if and only if a quadratic form, defined in terms of the angles between the reflecting hyperplanes, is positive definite (as opposed to “positive semidefinite” in the case of an infinite reflection group).

Next, in §5, he defines a “Vektordiagramm” which is an axiomatic description of a slight generalization of the notion of a reduced root system. The main result here (Satz 9) is the enumeration of all possible Vektordiagramme. Then he describes necessary and sufficient axiomatic conditions for a Vektordiagramm to define a unique semisimple complex Lie algebra. This result is summarized in table (31), which lists all those Vektordiagramme corresponding to a complex semisimple Lie algebra. Their Weyl groups correspond precisely to the positive definite “Figuren” A, ..., G of Satz 8 except D_{(n)} for n = 5 or n ≥ 7.

In table (31) as well as elsewhere, Witt does not use the Killing-Cartan notation, therefore we give a translation table:

Killing: A_{n}  B_{n}  C_{n}  D_{n} E_{n=6,7,8} F_{4} G_{2}
Witt: A_{n}^{∗} C_{n}^{∗}, λ = 1 C_{n}^{∗}, λ = 2 B_{n}^{∗} E_{n=6,7,8}^{∗} F_{4}^{∗} D_{(6)}^{∗}

Satz 15 proves the existence of a semisimple Lie algebra of any Killing type under the assumption that the existence is known for small (≤ 4) rank. In §7 a case by case construction of all possible reduced root systems is given.

Remarks on reflection groups and semisimple Lie algebras

In 1947, E. B. Dynkin [12] published a proof of the enumeration of complex semisimple Lie algebras which is based, similarly as in [W], on the previous work of Weyl and v. d. Waerden, but which, in contrast to [W], does not use Coxeter’s results on reflection groups. Dynkin’s method was developed in a seminar of I. M. Gelfand in Moscow in 1944. In a footnote [12, p. 59] the author refers to [W], saying that this article came to his knowledge only after his own article [12] was submitted for publication. Coxeter diagrams and Dynkin diagrams resemble each other in a striking way. For example, in [15], Tits uses the terminology “Witt-Dynkin diagram”. However, despite the graphical similarity of the Coxeter diagrams used by Witt and the Dynkin diagrams, they have slightly distinct semantics. The Coxeter diagram describes the presentation of a reflection group (in our case: the Weyl group of a semisimple Lie algebra) in terms of generating reflections (cf. [W, §3, p. 298]). The Dynkin diagram [12, §7, Nr. 47] describes the metric properties (mutual angles as well as the lengths) of a set of simple roots [12, §6, Nr. 35] of the root system of some semisimple Lie algebra. The notion of simple roots seems to be due to Dynkin [l.c.]. The Lie algebras of Killing type B_{n} and C_{n} are an example of two algebras having the same Coxeter diagram, but different Dynkin diagrams. Witt uses a non-canonical parameter λ to distinguish these two cases (see the translation table above). In the context of Kac-Moody Lie algebras, the distinction between the two types of diagrams is even more striking: for example, to each of the Figuren R_{n} in [W, Satz 8] there correspond three distinct types of Kac-Moody Lie algebras, which are distiguished by Dynkin diagrams (see below for references). Nevertheless, Witt was the first who used diagrams in order to enumerate semisimple Lie algebras.

Witt’s article [W] has been quite influential. For example, the standard reference N. Bourbaki, Groupes et algèbres de Lie, ch. 4,5,6, [21], follows essentially the ideas and lines of Witt’s proof, starting with the discussion of Coxeter groups (= groups generated by reflections), then discussing Weyl groups and finally giving the classification of root systems. In particular, for the construction of the exceptional root systems Witt’s arguments are used. This part of [W] has explicitly been recognized in the literature to be very elegant (cf. [13, p. 1136]).

The simplical complex 𝔗 constructed in [W, Satz 2] occurs as the geometric realization of the complex describing an apartment of a building with Weyl group Γ. In the spherical case, that is, for finite reflection groups [W, Satz 2(f)], we obtain the complex 𝔗 as an apartment of the Tits building [26] of a split semisimple Lie group with Weyl group Γ. Similarly, in the case of a contractible complex corresponding to an infinite reflection group, 𝔗 appears as an apartment of the Bruhat-Tits building [25] of a split semisimple Lie group with affine Weyl group Γ over a totally disconnected local field (see below). So, in this sense, the complex 𝔗 can be considered as the first building appearing in the mathematical literature. Of course it is only a thin one.

The complex 𝔗 is often called “Coxeter complex”. However, when Tits discusses the notion of a Coxeter complex as a constituent of buildings in [26, 2.11, p. 18], he refers to [21, ch. 5, n^{∘} 4.3], saying: “… the construction is essentially due to E. Witt” (then referring to [W]). To me, it seems that this complex rather should be called “Witt complex” or “Witt-Coxeter complex”, since it obviously has been introduced by Witt (cf. Satz 2). The important role of Witt’s arguments in this context is explicitly acknowledged by Coxeter in [6, §1.2, p. 382f.].

It was already mentioned that Witt did not only give an enumeration of finite, but also of infinite Coxeter groups. As we know today, most of the latter can appear as the so-called affine Weyl group associated to a reduced root system (cf. [21, ch. 6, §2]). This notion later became meaningful in various contexts, as I will briefly describe now.

For example, there is an interesting article of Stiefel [11], written slightly after Witt’s. It also gives an enumeration of semisimple Lie groups in terms of reflection groups. Stiefel, however, considers compact forms of Lie groups, and the reflection group he associates to each of them is its affine Weyl group, operating by affine transformations on some ℝ^{ℓ} as the simply connected covering of a maximal torus (as we would say today – Stiefel uses the term “toroid”) of the compact Lie group. Stiefel classifies semisimple compact Lie groups not only up to local isomorphy. He uses the operation of the affine Weyl group in order to distinguish groups of same isogeny type as well [11, §4]. At the end of his article (§5) he refers to the results of Coxeter and also to [W], and he expresses his hope that also his critical group Γ (the affine Weyl group) should be describable by means of “modern crystallography”. If he had looked into Satz 8 of [W] more carefully he probably would have found what he wanted!

Later, Iwahori and Matsumoto [18] discovered that the affine Weyl group occurs naturally for semisimple p-adic Lie groups and that it even shows up there in the context of so-called BN-pairs or Tits systems [21]; and this has been used, for example, to describe the conjugacy classes of maximal compact subgroups of semisimple p-adic Lie groups. On the basis of the results of Iwahori and Matsumoto the already mentioned contractible Bruhat-Tits buildings for reductive p-adic Lie groups were constructed (cf. [25]), which appeared in many applications in the theory of linear algebraic groups.

The “affine counterpart” of Witt’s enumeration of semisimple Lie algebras is the construction of the Kac-Moody Lie algebras [22, 23]. These are infinite dimensional Lie algebras, and an important class of them corresponds to Figuren P, …, W of Satz 8, that is, they have affine Weyl groups. The tables corresponding to Witt’s table (31) are given by [22, table 1,2,3, p.1291f] as well as by [23, p. 229]. The discovery of these algebras has opened a wide new field of research. In particular, corresponding Kac-Moody groups have been constructed and investigated [27].

Refined versions of the enumeration of semisimple Lie algebras have been given after Witt by many mathematicians. I mention the proof of Chevalley, who was able to normalize the structural constants N_{α,β} (cf. §6 of [W]) to have values in ℤ [14, Th. 1, p. 24], thereby constructing semisimple Lie algebras over ℤ for any reduced root system [14, §III, p. 32]. This made it possible to define the later so-called Chevalley groups as group schemes over ℤ and hence over any commutative ring [16, 17]. A further normalization was obtained by Tits in [19], who was able to resolve an ambiguity for the signs of the constants N_{α,β} which was left over in the construction of Chevalley.

The following chronological table shows the article [W] of Witt in the middle of a centenary approach of prominent mathematicians to understand and develop an exiting and influential area of mathematics.

Enumeration of enumerators of semisimple Lie algebras and related structures

1888/9 W. Killing Classification of semisimple Lie groups,

root systems, invention of the notion halbeinfach (semisimple)

1894 E. Cartan Internal structure of semisimple Lie groups

and algebras

1925 H. Weyl Representations of Lie groups,

“Weyl group” generated by reflections

1933 B. L. v. d. Waerden Classification of semisimple Lie algebras reduced to classification of root systems

1934 H. S. M. Coxeter Enumeration of reflection groups

1941 E. Witt Enumeration of semisimple Lie algebras by using Coxeter’s classification of reflection groups, invention of the “Witt(-Coxeter) complex”

1941/2 E. Stiefel Enumeration of compact Lie groups by

means of affine Weyl groups

1947 E. D. Dynkin Enumeration of semisimple Lie algebras by

means of “Dynkin diagrams”

1955 C. Chevalley “Integralization” of semisimple Lie algebras, normalization of the structural constants of semisimple Lie algebras

1961 C. Chevalley “Chevalley groups” as group schemes over

ℤ

1966 J. Tits Normalization of the structural constants

revisited

1968 V. G. Kac, R. V. Moody Independent description and enumeration of infinite dimensional “Kac-Moody” Lie algebras

1972 F. Bruhat, J. Tits Enumeration of affine buildings
1974 J. Tits  Enumeration of spherical buildings

1987 J. Tits Enumeration of Kac-Moody groups over arbitrary fields

References

[W] E. Witt, Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe, Abhandlungen aus dem mathematischen Seminar der Hansischen Universität 14 (1941), 289-322.

The first eight articles mentioned below are the content of the Literaturverzeichnis in [W, p. 290], where they appear under the same numbers.

[1] E. Artin, Die Aufzählung aller einfachen kontinuierlichen Gruppen (Göttinger Vorträge 13./15. Juli 1933).

[2] E. Cartan, Sur la structure des groupes de transformations finis et continus (Thèse, Paris 1894), in: Œvres complètes I, Gauthier-Villars, Paris (1952), 137-287.

[3] H. S. M. Coxeter, The polytopes with regular-prismatic vertex figures, Proc. Lond. Math. Soc. (2) 34 (1932), 126-189.

[4] H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. 35 (1934), 588-621.

[5] H. S. M. Coxeter, The complete enumeration of finite groups of the form R_{i}^{2} = (R_{i}R_{j})^{k_{ij}} = 1, J. Lond. Math. Soc. 10 (1935), 21-25.

[6] H. S. M. Coxeter, Regular and semi-regular polytopes, Math. Z. 46 (1940) 380-407.

[7] B. L. v. d. Waerden, Die Klassifikation der einfachen Lieschen Gruppen, Math. Z. 37 (1933), 446-462.

[8] H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I Math. Z. 23 (1925), 271-309, II ibid. 24 (1925), 328-376, III ibid. 24 (1925), 377-395.

[9] W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen: I Math. Ann. 31, (1888), 252-290 II ibid. 33, (1889), 1-48 III ibid. 34, (1889), 57-122 IV ibid. 36, (1890), 161-189.

[10] J. A. Schouten, Vorlesung über die Theorie der halbeinfachen kontinuierlichen Gruppen, mimeographed notes, Leiden 1926-27

[11] E. Stiefel, Über eine Beziehung zwischen geschlossenen Lie’schen Gruppen und diskontinuierlichen Bewegungsgruppen euklidischer Räume und ihre Anwendung auf die Aufzählung der einfachen Lie’schen Gruppen, Comm. Math. Helv. 14 (1941-42), 350-380.

[12] E. B. Dynkin, The structure of semi-simple Lie algebras, Uspehi Mat. Nauk 2 4(20) (1947), 59-127, (Transl. Amer. Math. Soc. 17).

[13] C. Chevalley, Sur la classification des algèbres de Lie simples et de leurs représentations, C. R. Acad. Sci. 227 (1948), 1136-1138.

[14] C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14-66.

[15] J. Tits, Les groupes de Lie exceptionnels et leur interprétation géométrique, Bull. Soc. Math. Belg. 8 (1956), translated into German and reprinted in: H. Freudenthal, Raumtheorie, Wiss. Buchges., Darmstadt (1978) 250-282.

[16] T. Ono, Sur les groupes de Chevalley, J. Math. Soc. Japan 10 (1958), 307-313.

[17] C. Chevalley, Certains schémas de groupes semi-simples, Sém. Bourbaki 219 (1960/1).

[18] N. Iwahori, H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of 𝔭-adic Chevalley groups, Publ. math. IHES 25 (1965), 5-48.

[19] J. Tits, Sur les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples, Publ. math. IHES 31 (1966), 21-58.

[20] J. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Math. 40, Springer, Berlin (1967).

[21] N. Bourbaki, Groupes et algèbres de Lie, ch. 4,5,6, Éléments de Mathématique XXXIV, Hermann, Paris (1968).

[22] V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. USSR-IZV. 2, n^{∘} 6 (1968), 1271-1311.

[23] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211-230.

[24] H. S. M. Coxeter, W. O. J. Moser, Generators and relations for discrete groups, Erg. Math. 14, 3^{rd} ed., Springer, Berlin (1972).

[25] F. Bruhat, J. Tits, Groupes reductifs sur une corps local I, Publ. math. IHES 41 (1972), 5-252.

[26] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. 386, Springer, Berlin (1974).

[27] J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), 542-573.