Lie Groups and Lie Algebras, Part II, Chapters 4-6

312 Pages·2002·148.47 MB·English

by  Nicolas Bourbaki

. ELEMENTS OF MATHEMATICS
J -
Springer
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Heidelberg
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NICOLAS BOURBAKI
,r'"'
~/
ELEMENTS OF MATHEMATICS
/lie
Groups and
Lie Algebras .
/ /
Chapters 4-6
~
Springer
Originally published as
GROUPES ET ALGEBRES DE LIE
·7
Hermann, Paris, 1968
Translator
Andrew Pressley
Department of Mathematics
King's College London
Strand, London WC2R 2LS
United Kingdom
e-mail: [email protected]
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Einheitssacht.: Elements de mathematique <engl.>
Lie groups and Lie algebras
Chapters 4/6.-2002
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INTRODlJCTION TO CHAPTERS IV, V
AND VI
The study of semi-simple (analytic or algebraic) groups and their Lie algebras
leads to the consideration of root systems, Coxeter groups and Tits systems.
Chapters IV, V and VI are devoted to these structures.
To orient the reader, we give several examples below.
I. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra
of g1. A root of g with respect to f) is a non-zero linear form a on f) such
that there exists a non-zero element x of g with [h, x] = a(h)x for all hE f).
The~e roots form a reduced root system R in the vector space f)* dual to
f). Glving R determines g up to isomorphism and every reduced root system
is isomorphic to a root system obtained in this way. An automorphism of g
leaving f) stable defines an automorphism of f)* leaving R invariant, and every
automorphism of R is obtained in this way. The Weyl group of R consists of
all the automorphisms of f)* defined by the inner automorphisms of g leaving
f) stable; this is a Coxeter group.
Let G be a connected complex Lie group with Lie algebra g, and let r
be the subgroup of G consisting of the elements h such that expc ( 21rih) = 1.
Let R- be the root system in f) inverse to R, let Q(R-) be the subgroup of f)
generated by R- and let P(R-) be the subgroup associated to the subgroup
Q(R) of f)* generated by R (i.e. the set of h E f) such that >.(h) is an integer
r
for all ).. E Q(R)). Then P(R~) :J :J Q(R~). Moreover, the centre of G
r
is canonically isomorphic to P(R-)/ and the fundamental group of G to
r r
E /Q(R-). In particular, is equal to P(R~) if G is the adjoint group and
is equal to Q(R~) if G is simply-connected. Finally, the weights of the finite
dimensional linear representations of G are the elements of the subgroup of
r.
f)* associated to
II. Let G be a semi-simple connected compact real Lie group, and let g be its
Lie algebra. Let T be a maximal torus of G, with Lie algebra t, and let X be
the group of characters of T. Let R be the set of non-zero elements a of X
such that there exists a non-zero element x of g with (Ad t).x = a(t)x for all
t E T. Identify X with a lattice in the real vector space V =X @z R; then R
. is a reduced root system in V. Let N be the normaliser ofT in G; the ac~ion
1 In this Introduction, we use freely the traditional terminology as well as the
· notions defined in Chapters IV, V and VI.
VI TO CHAPTERS IV, V AND VI
of N on T defines an isomorphism from the group N /T to the Weyl group of
R. We have P(R) ::) X::) Q(R); moreover, X= P(R) if G is simply-connected
and X= Q(R) if the centre of G reduces to the identity element.
The complexified Lie algebra fl(e) of g is semi-simple and tee) is a Cartan
subalgebra of it. There exists a canonical isomorphism from V(e) to the dual
of tee) that transforms R into the root system of flee) with respect to tee).
III. Let G be a connected semi-simple algebraic group over a commutative
field k. Let T be a maximal element of the set of tori of G split over k and
let X be the group of characters of T (the homomorphisms from T to the
multiplicative group). We identify X with a lattice in the real vector space
V = X @z R. The roots of G with respect to T are the non-zero elements a of
X such that there exists a non-zero element x of the Lie algebra g of G with
(Ad t).x = a(t)x for all t E T. This gives a root system R in V, which is not
necessarily reduced. Let N be the normaliser and Z the centraliser of T in G
and let N(k) and Z(k) be their groups of rational points over k. The action
of N(k) on T defines an isomorphism from N(k)/Z(k) to the Weyl group of
R.
Let U be a maximal element of the set of unipotent subgroups of G, de
fined over k and normalised by Z. Put P = Z.U. Then P(k) = Z(k).U(k) and
P(k) n N(k) = Z(k). Moreover, there exists a basis (ar, ... , an) of R such
that the weights ofT in U are the positive roots of R for this basis; if S de
notes the set of elements of N(k)/Z(k) that correspond, via the isomorphism
defined above, to the symmetries sa.i E W(R) associated to the roots o:i, the
quadruple (G(k), P(k), N(k), S) is a Tits system.
IV. In the theory of semi-simple algebraic groups over a local field, Tits
systems are encountered whose Weyl group is the affine Weyl group-of a
root system. For example, let G = SL(n + 1, QP) (with n 2:: 1). Let B be
the group of matrices (aij) E SL(n + 1, Zp) such that a~j E pZp fori < j,
and let N be the subgroup of G consisting of the matric~s having only one
non-zero element in each row and column. Then there exists a subset S of
N/(B n N) such that the quadruple (G, B, N, S) is a Tits system. The group
W = N / (B n N) is the affine Weyl group of a root system of type An; this is
an infinite Coxeter group.
Numerous conversations with J. Tits have been of invaluable assistance to us
in the preparation of these chapters. We thank him very cordially.
CONTENTS
INTRODUCTION TO CHAPTERS IV, V AND VI . . . . . . . . V
CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
CHAPTER IV COXETER GROUPS AND TITS SYSTEMS
§ l.:"Coxeter Groups ....................................... :. 1
1. Length and reduced decompositions .................... . 1
2. Dihedral groups ..................................... . 2
3. First properties of Coxeter groups ..................... . 4
4. Reduced decompositions in a Coxeter group ............. . 5
5. The exchange condition .............................. . 7
6. Characterisation of Coxeter groups ..................... . 10
\. ../ 7. Families of partitions .......... ....................... . 10
8. Subgroups of Coxeter groups .......................... . 12
9. Coxeter matrices and Coxeter graphs ................... . 13
§ 2. Tits Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Definitions and first properties . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3. Decomposition of G into double cosets . . . . . . . . . . . . . . . . . . 18
. 4. Relations with Coxeter systems . . . . . . . . . . . . . . . . . . . . . . . . . 19
5. Subgroups of G containing B . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6; Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7. The simplicity theorem . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 23
'··,·':>:- ;,A:p•peud.ix. Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
The connected components of a graph . . . . . . . . . . . . . . . . . . . 27
Forests and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
:-l:!lx.erc:ises for § 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
~·~'--'-" for § 2 • . • • • • • • . . . . . • • • • • • • . . • • • . • • • • • • . . . . • • • • • • • • • . . 44
V.LOvO
VIII CONTENTS
CHAPTER V GROUPS GENERATED BY REFLECTIONS ,
§ 1. Hyperplanes, chambers and facets . . . . . . . . . . . . . . . . . . . . . . 61
1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2. Facets ..................................... ~ . . . . . . . . . 62
3. Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4. Walls and faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5. Intersecting hyperplanes .................. _: ~ . . . . . . . . . . . 67
6. Simplicial cones and simplices ............. -. . . . . . . . . . . . . 68
§ 2. Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1. Pseudo-reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3. Orthogonal reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4. Orthogonal reflections in a euclidean affine space . . . . . . . . . 73
5. Complements on plane rotations . . . . . . . . . . . . . . . . . . . . . . . . 74
§ 3. Groups of displacements generated by reflection_s . . . . . . . 76
1. Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2. Relation with Coxeter systems . . . . . . . . . . . . . . . . . . . . . . . . . 78
3. Fundamental domain, stabilisers . . . . . . . . . . . . . . . . . . . . . . . . 79
4. Coxeter matrix and Coxeter graph of W . . . . . . . . . . . . . . . . . 81
5. Systems of vectors with negative scalar products . . . . . . . . . . 82
6. Finiteness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7. Decomposition of the linear representation of WonT . . . . . 86
8. Product decomposition of the affine space E . . . . . . . . . . . . . . 88
9. The structure of chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10. Special points .....' ...................... ·: . . . . . . . . . . . 91
§ 4. The geometric representation of a Coxeter group . . . . . . . 94
1. The form associated to a Coxeter group . . . . . . . . . . . . . . . . . 94
2. The plane Es,s' and the group generated by a- and O"s' 95
8
3. The group and representation associated to a Coxeter
matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4. The contragredient representation . . . . . . . . . . . . . . . . . . . . . . 97
5. Proof of lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6. The fundamental domain of W in the union of the
chambers ............................................ · 101
7. Irreducibility of the geometric representation of a Coxeter
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8. Finiteness criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9. The case in which BM is positive and degenerate . . . . . . . . . . 105
CONTENTS IX
§ 5. Invariants in the symmetric algebra . . . . . . . . . . . . . . . . . . . . . 108
1. Poincare series of graded algebras . . . . . . . . . . . . . . . . . . . . . . . 108
2. Invariants of a finite linear group: modular properties 110
3. Invariants of a finite linear group: ring-theoretic properties . 112
4. Anti-invariant elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5. Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
§ 6. The Coxeter transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1. Definition of Coxeter transformations . . . . . . . . . . . . . . . . . . . 121
2. Eigenvalues of a Coxeter transformation: exponents . . . . . . . 122
Appendix: Complements on linear representations . . . . . . . . . . 129
Exercises for § 2. 133
Exercises for § 3. 134
Exercises for § 4. 137
Exercises for § 5. 144
Exercises for § 6. 150
CHAPTER VI ROOT SYSTEMS
§ 1. Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
1. Definition of a root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
2; Direct sum of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3. Relation between two roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4. Reduced root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5. Chambers and bases of root systems . . . . . . . . . . . . . . . . . . . . 166
6. Positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7. Closed sets of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8. Highest root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9. Weights, radical weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10. Fundamental weights, dominant weights . . . . . . . . . . . . . . . . . 180
11. Coxeter transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
12. Canonical bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Affine Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
1. _ Affine Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2. Weights and special weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
The normaliser of W a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Application: order of the Weyl group . . . . . . . . . . . . . . . . . . . . 190
Root systems and groups generated by reflections . . . . . . . . . 191
X CONTENTS
§ 3. Exponential invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
1. The group algebra of a free abelian group . . . . . . . . . . . . . . . . 194
2. Case of the group of weights: maximal terms . . . . . . . . . . . . . 195
3. Anti-invariant elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4. Invariant elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
§ 4. Classification of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1. Finite Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2. Dynkin graphs ........................... --:........... 207
3. Affine Weyl group and completed Dynkin graph . . . . . . . . . . 210
4. Preliminaries to the construction of root systems . . . . . . . . . 212
5. Systems of type Bz (l 2:: 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6. Systems of type Cz (l 2:: 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7. Systems of type Az (l 2:: 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8. Systems of type Dz (l 2:: 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9. System of type F 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10. System of type E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
11. System of type E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
12. System of type E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
13. System of type G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
14. Irreducible non-reduced root systems . . . . . . . . . . . . . . . . . . . . 233
Exercises for § 1. 235
Exercises for § 2. 240
Exercises for § 3. 241
Exercises for § 4. 242
HISTORICAL NOTE (Chapters IV, V and VI) . . . . . . . . . . . . 249
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . 255
INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
INDEX OF TERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
PLATE I. Systems of type A (l 2:: 1) . . . . . . . . . . . . . . . . . . . . . . . . 265
1
PLATE II. Systems of type Bz (l 2:: 2) . . . . . . . . . .. . . . . . . . . . . . . . 267
PLATE III. Systems of type C (l 2:: 2) . . . . . . . . . . . . . . . . . . . . . . 269
1
PLATE IV. Systems of type Dz ( l 2:: 3) 271
PLATE V. System of type E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
PLATE VI. System of type E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
PLATE VII. System of type E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283