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Afterword (1994)

Each reprinting of this text has given me an opportunity to correct mis­prints and errors pointed out by patient readers. The most substantial change was the addition of the appendix to §24 at the time of the second printing. If I were starting today from scratch, I would certainly do some things-large and small-differently. In the area of notation, for example, p is now more commonly used than (j to denote the half-sum of positive roots. But with many copies of the book already in circulation, I have been reluctant to disturb the existing format.

While the structure theory developed in Chapters I-V has evolved very little over the past 25 years, there has been an explosion of new work in representation theory. The foundations laid here in Chapter VI are still valid, being aimed primarily at the classical finite dimensional theory of Cartan and Weyl. However, some of the ad hoc terminology and notation I introduced have long since been replaced by other conventions in most of the literature: In place of "standard cyclic modules" one now speaks of "highest weight modules", the universal ones being called "Verma modules". Verma modules are usually denoted M(A) rather than Z(A), while the irreducible quotient is labelled L(A). Of course, Lie theory is a sprawling subject, with many conflict­ing notational schemes (especially for root systems). So the student has to be somewhat flexible in any case when approaching the literature.

The present text contains much of the standard core of semisimple Lie theory, in a purely algebraic setting. This theory-especially the classification of simple Lie algebras by Dynkin diagrams-is beautiful in its own right, whatever one's ulterior motive for studying it. But readers should be aware of the far-reaching developments of recent decades that rely in some way on this core. While it is impossible in a page or two to survey these developments adequately, a quick overview may be useful. References below are mainly to books rather than to the multitude of original articles; the latter are well documented in the annual subject index of Mathematical Reviews. With due apology for omissions, here are some of the subjects most closely related to semisimple Lie algebras:

• The BGG category @. This consists of finitely-generated weight modules on which a fixed Borel subalgebra acts locally finitely. It includes Verma modules and irreducible highest weight modules L(A) for arbitrary A, as well as projective and injective objects. Besides the BGG resolution of a finite dimensional L(A) by Verma modules, which makes more concrete the derivation of Weyl's character formula presented here, one encounters BGG reciprocity along with the Jantzen filtration and sum formula: see J. C. Jantzen, Moduln mit einem hOchsten Gewicht, Lect. Notes in Mathe­matics 750, Springer-Verlag, 1979.

167

168 Afterword (1994)

• Kazhdan-Lusztig conjectures. A conjectured character formula for all L(A) appeared in the seminal paper by D. A. Kazhdan and G. Lusztig, Represen­tations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. This formula was quickly proved (independently) by Beilinson­Bernstein and by Brylinski-Kashiwara, using a dazzling array of tech­niques. The Hecke algebr~ approach has become extremely influential in several kinds of representation theory.

• Primitive ideals in enveloping algebras. Combining noncommutative ring theory and algebraic geometry with the representation theory of semisimple Lie algebras yields deep results on the structure of universal enveloping algebras. See J. Dixmier, Algebres enveloppantes, Gauthier-Villars, 1974, and J. C. Jantzen, Einhiillende Algebren halbeinfacher Lie-Algebren, Springer-Verlag, 1983.

• Lie group representations. Lie algebra techniques indicated above have led to decisive progress in many areas of the representation theory of semi­simple (or reductive) Lie groups. See for example D. A. Vogan, Jr., Represen­tations oj Real Reductive Lie Groups, Birkhiiuser, 1981.

• Representations oj algebraic groups. Much of the theory of semisimple Lie algebras can be adapted to semisimple algebraic groups in arbitrary charac­teristic. Representations in characteristic p are somewhat like infinite di­mensional representations in characteristic o. See J. C. Jantzen, Represen­tations oj Algebraic Groups, Academic Press, 1987.

• Finite groups oj Lie type. Lie theory is essential to understanding the structure of these groups, as well as their ordinary and modular representa­tions. See R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, 1985.

• Kac-Moody Lie algebras and vertex operators. The Serre relations of §18 lead to new classes of infinite dimensional Lie algebras, when the Cart an matrix is replaced by a "generalized Cartan matrix". These Kac-Moody Lie algebras and their representations interact deeply with mathematical physics, combinatorics, modular functions, etc. See V. G. Kac, Infinite Dimensional Lie Algebras, 3rd edition, Cambridge University Press, 1990, and I. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988.

• Quantum groups. Since the pioneering work of Drinfeld and Jimbo in the mid-eighties, quantized enveloping algebras have become ubiquitous in mathematics and mathematical physics. See G. Lusztig, Introduction to Quantum Groups, Birkhiiuser, 1993, and J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge University Press, 1992.

• Combinatorics, geometry, etc. Apart from their connection with Lie algebras, root systems and root lattices along with related Coxeter groups such as Weyl groups play an essential role in many areas: Macdonald formulas, quivers and representations of finite dimensional algebras, singu­larities, crystals and quasi-crystals, etc. See for example 1. H. Conway, N. J. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, 1993.

Index of Terminology

a-string through fJ 39,45 Coxeter graph 56 ex-string through p. 1I4 abelian Lie algebra 4

degree 139 abstract Jordan decomposition 24

derivation 4 adjoint Chevalley group 150

derived algebra 6 adjoint representation 4

derived series 10 admissible lattice 159

descending central series II ad-nilpotent 12 ad-semisimple 24 diagonal automorphism 87

affine n-space 132 diagonal matrices 3

algebra 4 diagram automorphism 66

associative bilinear form 21 direct sum of Lie algebras 22

automorphism 8 dominant weight 67 dominant integral linear function 112 dual module 26

base of root system 47 dual root system 43 Borel subalgebra 83 Dynkin diagram 57 bracket 1,2

Engel subalgebra 79 canonical map 7 Engel's Theorem 12 Cartan decomposition 35 epimorphism 7 Cartan integer 39,55 equivalent representations 25 Cartan matrix 55 exceptional Lie algebras 102 Cartan subalgebra (CSA) 80 Cartan's Criterion 20

faithful representation 27 Casimir element 27, 1I8 Cayley algebra 104 flag 13

formal character 124 center (of Lie algebra) 6

free Lie algebra 94 center (of universal enveloping

Freudenthal's formula 122 algebra) 128

fundamental domain 52 centralizer 7

fundamental dominant weight 67 character 129 character (formal) 124

fundamental group 68

Chevalley algebra 149 fundamental Weyl chamber 49

Chevalley basis 147 Chevalley group 150, 163 general linear algebra 2 Chevalley's Theorem 127 general linear group 2 classical Lie algebra 2 generators and relations 95 Clebsch-Gordan formula 126 graph automorphism 66,87 closed set of roots 87, 154 group ring 124 commutator 1 completely reducible module 25 Harish-Chandra's Theorem 130 contragredient module 26 height 47 convolution 135 highest weight 32,70, 108

169

170 Index of Terminology

homogeneous symmetric tensor 90 nilpotent Lie algebra 11 homomorphism (of Lie algebras) 7 nilpotent part 17,24 homomorphism (of L-modules) 25 nondegenerate bilinear form 22 hyperplane 42 non-reduced root system 66

normalizer 7

ideal 6 octonion algebra 104 induced module 109 orthogonal algebra 3 inner automorphism 9 orthogonal matrix 10 inner derivation 4 outer derivation 4 integral linear function 112 invariant polynomial function 126 parabolic subalgebra 88 inverse root system 43 partition function 136 irreducible module 25 PBW basis 92 irreducible root system 52

Poincare-Birkhoff-Witt Theorem 92 irreducible set 133 polynomial function 126, 133 isomorphism (of Lie algebras) 1 positive root 47 isomorphism (of L-modules) 25 isomorphism (of root systems) 43

quotient Lie algebra 7

Jacobi identity 1 Jordan-Chevalley decomposition 17 radical (of bilinear form) 22

radical (of Lie algebra) 11 rank (of Lie algebra) 86

Killing form 21 rank (of root system) 43 Kostant function 136 reduced 51 Kostant's formula 138 reductive Lie algebra 30, 102 Kostant's Theorem 156 reflecting hyperplane 42

reflection 42

lattice 64, 157 regular 48

length (in Weyl group) 51 regular semisimple element 80

Lie algebra 1 representation 8

Lie's Theorem 16 root 35

linear Lie algebra 2 root lattice 67

linked weights 129 root space decomposition 35

locally nilpotent 99 root system 42

long root 53 lower central series 11 saturated set of weights 70

scalar matrices 5

maximal toral subalgebra 35 Schur's Lemma 26 self-normalizing subalgebra 7 maximal vector 32, 108 semisimple endomorphism 17 minimal weight 72 semisimple Lie algebra 11 module (for Lie algebra) 25 semisimple part 17,24 monomorphism 7 Serre's Theorem 99 multiplicity of weight 117 sbort root 53 simple Lie algebra 6

negative root 47 simple reflection 51 nilpotelJt endomorphism 8 simple root 47

Index of Terminology 171 singular 48 toral subalgebra 35 skew-symmetric tensor 117 trace 2 solvable Lie algebra 10 trace polynomial 128 special linear algebra 2 special linear group 2 standard Borel subalgebra 84 universal Casimir element 118

standard cyclic module 108 universal Chevalley group 161

standard parabolic subalgebra 88 universal enveloping algebra 90

standard set of generators 74 upper triangular matrices 3

Steinberg'S formula 141 strictly upper triangular matrices 3 weight 31,67, 107 strongly ad-nilpotent 82 weight lattice 67 strongly dominant weight 67 weight space 31, 107 structure constants 4 Weyl chamber 49 subalgebra (of Lie algebra) 1 Weyl function 136 support 135 Weyl group 43 symmetric algebra 89 Weyl's formulas 139 symmetric tensor 90 Weyl's Theorem (complete symplectic algebra 3 reducibility) 28

tensor algebra 89 Zariski topology 133 tensor product of modules 26

Index of Symbols

[xy] 1 :> 47 At 2 11>+ 47 Ct 2 11>- 47 gI (n, F) 2 ~(,,) 48 gI(V) 2 [(,,) 49 End V 2 [(~) 49 GL(V) 2 I) 50, 70 sl( V) 2 tea) 51 sl(n, F) 2 n(a) 51 Tr 2 sn(a) 54, 136 SL(V) 2 F4 58 St 3 E6 , E7, Es 58 Dt 3 r 65 sp(V) 3 A 67, 112 sp(n, F) 3 Ar 67 o(V) 3 A+ 67, 112 o(n, F) 3 .\. 67 ten, F) 3 r(L) 77,87 n(n, F) 3 LQ(ad x) 78 ben, F) 3 CSA 80 Der 111 4 .,v(L) 82 Z(L) 6 G(L) 82 NL(K) 7 G(L; K) 82 CL(X) 7 B(~) 84 Int L 9 N(~) 84 L(I) 10 X(V) 89 LI 11 6(V) 89 RadL 11 .9m 90 K(X, y) 21 6(V) 90 ctp 27 U(L) 91 II> 35,42 PSW 92 ttl 37 (£ 104 h,. 37 [0 105 S,. 38 VA 107 E 40,42 Z('\) 110 <cx,{3) 42 J(.\) 110 a,. 42 Y('\) 110 P,. 42 V('\) 110 "Ir 43 II(V) 113 II>Y 43 11(,\) 113 Y 43 mAef') 117 IX

Gz 44 m(f') 118 ~ 47 cL 118 ht ex 47 Z[A] 124

172

Index of Symbols 173

e('\) 124 ID1 .. 137

ch .. 124 (}(,\) 137

chI' 124 deg (,\) 139 Ip{V) 126 cf%{J 145 Ip{H)"Ir 127 L(Z) 149 Ip(L)G 127 L(K) 149

3 128 G(K) 150

x .. 129 U(L)z 156 ,\,....,,.,. 129 H(Z) 159 fJt 133 Ly 159

1: 13S Mmln 161

f.g 13S Mmax 162

E .. 13S Gy(K) 162

p('\) 13S L,,(K) 162

q('\) 136

Graduate Texts in Mathematics (continued from page ii)

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