Index of Notation

1.1.1. (/,•); vi/ 1.1.2. Pi (for P E Q(v)); vv

(for i/ G Z[/]); tr v

1.2.1. 'f; 'f„; \x\ for x G 'f; 0,; N[J] 1.2.2. r —• 'f <g> 'f 1.2.3. (,) on 'f 1.2.4. J 1.2.5. f;f„; (,)on f;

|x| for x G f 1.2.6. r : f —» f <8> f 1.2.7. a on 'f; 1.2.9. cr on f 1.2.10. ~ on Q(v), on 'f;

f : 'f — W f , {,} on 'f 1.2.12. ~ on f; f : f — f(g>f;

{,} onf 1.2.13. r i ; ir 1.3.1. A; [?] 1.3.3. [n]; [n]!

1.4.1. 0\p)

1.4.7. ^f , ^ f , 2.1.1. h(ij); W 2.1.2. i(u>) 2.1.3. w0 2.2.1. V; X; (,); i ^ z; i i' 2.2.3. A < A'

2.3.1. 2p\ n 2.3.2. p(i) 3.1.1. 'U; 'U+; 'U" ;

U; U+; U"; Ei] Fi\ K F(p). f ( P ) . ' 1 i ' K^ Ku\ x x+; x * a:-

3.1.3. u; on U; cr on U 3.1.4. A o n U / U 3.1.12. " on U

3.1.13. ^U 3.2.4. U° 3.2.6. U+; U" 3.3.1. 5, 5' 3.4.1. C 3.4.4. wAf 3.4.5. Mx 3.4.7. 3.5.1. C' 3.5.5. X+ 3.5.6. AA 3.5.7. -Aa; *?a; 4.1.1. A 4.1.2. 9; e„ 4.1.4. {n}i 4.2.i. e 1 2 , e 2 3 , e 1 3

5.1.1. 5.2.1. : M M 5.3.1. LI; L'l 6.1.7. ttE : M M 7.1.1. /ij;n?m;e 3:1 /n,m;e 8.1.1. V(X)\ D(K); HnK; K\j] 8.1.2. M(AT); -M(X)[n]; HnK; r<n; 8.1.7. A4G(X); M g P O H ; A * 8.1.9. T>j(X,G\AyB) 8.1.11. CT\ $(K) 9.1.2. V; V„; G v ; E v 9.1.3. * , i/, ^Fi/j Lu 9.1.3. TV; Qv 9.2.1. PT,w; QT.W 9.2.5. IndT,w;

~ v nesT ^yj Ind^ w

9.2.10. Res^ yv 9.3.1. PV;I';>7; 7V,I';<y 11.1.5. O 12.1.2. V°; V°; Qv; /C(QV); £(7V) 12.1.3. O' = 0{v,v~l] 12.1.5. ind^ w ;

G. Lusztig, Introduction to Quantum Groups, Modern Birkhäuser Classics,

339DOI 10.1007/978-0-8176-4717-9, © Springer Science+Business Media, LLC 2010

340 Index of Notation

13.2.1. ^k 13.2.6. k 14.2.1. A 14.2.3. B 14.2.5. BV 14.3.1. 5i;>n; '£<;>„;

Bi-nl aBi,n 14.4.2. B; B„ 14.4.6. Bj;n; ffBj;n 14.4.11. B(A); B ( A a ) 15.1.2. 11 16.1.1. Vi 16.1.3. <i>i\ li 16.1.4. Fi\ Ei 16.2.3. A(Z); A 17.3.3. C(f) 18.1.1. (A)„ 18.1.2. B{A) 18.1.5. £(f)„ 18.1.8. L(A); L'(A) 19.1.1. p : U —• U 19.1.2. ( , ) on A 19.3.1. ^Aa 19.3.4. " : Aa - Aa

23.1.1. a'UA"! U; 1a 23.1.2. U(^) 23.1.8. ~ : U U 23.3.3. P(C,A,A'); AP{T,A,A')

24.3.3. hOx^'K 25.2.1. 60c6" 25.2.4. B

25.4.1. mcab>, mf

26.1.2. ( , ) on U 27.1.1. M[A]; M[> A]; M[> A] 27.2.1. £[A] 27.2.3. B\A]w; B[A]io

27.2.5. M* 27.3.2. hOb^ 29.1.1. B[Ai] 29.1.2. U[> Ai]; U[> Ai] 31.1.1. (j>: A-+ R\ j?U; Rf 31.1.5. RC 31.2.4. RC' 31.1.13. RMX 31.2.1. RAX, 31.3.2. RLX 32.1.5. /T^m.M' 33.1.1. Rf 35.1.3. A' 35.4.1. f 36.1.2. flf 36.2.1. RU 36.2.5. 37.1.2. T'i e,Tl[e 37.2.1. Xn rn^e\

2/n,ra;e> 2/n,m;e 38.1.1. f[i]; CTf[i] 38.1.3. g : f[i] ^[i] 39.4.4. TweJTwe

41.1.1. T'i e,T['e : U - > U 42.1.8. H

Index of Terminology

adapted basis, 16.3.1 admissible

automorphism, 12.1.1 form, 16.2.2

almost orthonormal, 14.2.1 antipode, 3.3.2

skew, 3.3.2

based module, 27.1.2 basis at oo, 20.1.1 braid group, 2.1.1 Brauer's eentralizer algebra, 27.3.10

canonical basis of f, 14.4.6 of A a , 14.4.12 of " A a (g) AA/, 24.3.4 of U , 25.2.4 signed...of f, 14.2.4 topological...of (U~"<g) U + y, 30.1.4

Cartan datum, 1.1.1 symmetric, irreducible, without

odd cycles, of finite type, of infinite type, 2.1.3

cell (two-sided, left, right), 29.4.1

coinvariants, 27.2.5 comultiplication 3.1.4 coordinate algebra

(quantum), 29.5.1 coroot, 2.3.1

discrete subset, 9.1.3 dominant, 3.5.5

equivariant perverse sheaf, 8.1.7 semisimple complex, 8.1.8

flag, 9.1.3 Fourier-Deligne transform, 8.1.11 Frobenius homomorphism

(quantum), 35.1.10

graph, 9.1.1

hexagon property, 32.2.4 highest weight module, 31.3.1

induction, 9.2.5 integrable object, 3.5.1

Kashiwara's operators, 16.1.4

left graph on B , 14.4.7 length of w, 2.1.2 linear category, 11.1.1

morphism of root data, 2.2.2

orientation, 9.1.1

periodic functor, 11.1.2 perverse sheaf, 8.1.2

quasi-classical case, 33.2.1 quasi-7£-matrix, 4.1.4

reduced expression, 2.1.2 restriction, 9.2.10 root datum, 2.2.1

y-regular..., X-regular..., simply connected..., adjoint..., 2.2.2

semisimple complex, 8.1.3 Serre relation

(quantum), 1.4.3 signed basis, 12.6.4 sink 9.3.4 structure constants 25.4.1 symmetries, 5.2.1

traceless, 11.1.4

triangular decomposition, 3.2, 23.2.1

unital module, 23.1.4

Verma module, 3.4.5, quantum .... identity, 39.3.7

weight space, 3.4.1 Weyl group, 2.1.1

Comments added in the second printing

1. Let M G C' and let i G / , e = ±1. We define two Q(v)-linear maps S t , e , S ? e : M ^ M by

a,6,c

a,b,c

where a, b, c run over N; although the sums are infinite, on any given vector in M, all but finitely many terms in either sum act as zero.

Several readers have asked me about the relationship between these op-erators (which appeared in [3, 3.1]) and the operators T!e,T"e : M —• M in 5.2.1. The relationship is as follows:

T>e e Diye ~ 1i,eIxi > Di,e ~ 1i,eJ^i •

To prove this, we may replace C by C[ and we may assume that M is a simple object of C-. Then the desired identities are checked by calculations similar to those in 5.2.2.

It follows that the braid group relations 39.4.3(c),(d) remain valid if T ' , T " are replaced throughout by S',S".

2. In the last sentence on p. 183, it was stated that it should be possible to remove the signs ± in Corollary 26.3.2. This is indeed true, as was shown by Kashiwara [2].

3. The question raised in the last sentence of 40.2.5 has been answered by J. Beck [1] in the untwisted case.

REFERENCES

[1] J. Beck, Convex bases of PBW type for quantum affine algebras, Communications in Math. Phys. (to appear).

[2] M. Kashiwara, Crystal base of modified enveloping algebras, Duke Math. J. 73 (1994), 383-414.

[3] G.Lusztig, Problems on canonical bases, Amer. Math. Soc., Proc. Symp. Pure Math. (1994).

Errata to Introduction to Quantum Groups (1994 printing)

p. 3 line 13: after "homomorphism," and before "then" insert: preserving Z[/]-gradings

p. 9, line -5: inside the last bracket [ ], replace o! by a" and t' by t".

p. 11, in the numerator of the first fraction on line -4, insert a factor vfvj

p. 12, in the numerator of the fraction on line -8, insert a factor vj

p. 13, just before 1.4.7, add the following text. Here is another proof of Proposition 1.4.3 which is not based on 1.4.5 but is based, instead, on the following statement which is easily verified. (a) For any k € I the left-hand side of the equality in 1.4.3 is in the kernel of r* : f —> f.

Note that 1.4.3 follows from (a) in view of 1.2.15(a).

p. 31, in 3.4.6 line 6: delete (a) (twice) and in 3.4.6 line -2, replace

p. 32 line 3.5.3(b): replace

E j F ? + Q t y F ? - 1 by E jF i 1 + Q{v)KiF^~l +

p. 32 line -1: add: See also 23.3.11.

p. 33, line 13, replace M\ by A^.

p. 40, line -3: replace n + 2m by n + 2m — 2.

p. 40, line -1: replace n — 2m by n — 2m + 2. p. 41 lines 5,6: replace If x 6 Mn(0), then by definition, (c — sn)x = 0 by

We have Mn(0) = 0.

p. 41 lines 9, 10, 11, 12: replace (four times) n + 2m by n + 2m — 2.

p. 42: delete lines 1,2.

p. 42 line 6: replace roman m,n by italic m,n.

p. 45: the equalities in line -2 of 5.2.6 should read

p. 45: the line "This follows immediately from the previous lemma" in 5.2.7 should read: This has been stated in the proof of the previous lemma.

p. 49, line -5: insert 2 in front of the second

p. 49, line -7: insert 2 in front of the second

p. 50, line 4: replace 2 by 4 in front of the second

p. 56, Lemma 7.1.4 should read: Assume that m = an + 1. We have

v71-1 _ v~n+1

p.f+ _ f+ p. — K 3 f+ ft" 3 f+ 3Jn,m\e Jn,m;ej-J ^», _ 01-1 •'n-l.mjl - 1 in-l ,m;-l• Vj Vj Vj V-

p. 57 line 3 and 4: the factors v®r(c m - m+1) should be be deleted.

p. 68 at the end of 9.1.2: insert the line:

"Here g\ : Vi —> Vi is the restriction of g"

p. 71 line -8: replace G xP F by G XQ F.

p. 72, line 7: replace T{ by TV p. 77 in 9.3.1 line -8: replace

Y(i) > 7(i) by ^ > yx

and 7r(i) > 7p) b y 7i > 7i-

p. 85, line 11: insert ) before =.

p. 87, lines 2,3 of 10.3.2: delete "and TV = 0 for all i' 6 I — I' "

p. 91, lines 7, 8, 12, 18: replace . . . (a*) by • • • 0 (a*) (in five places).

p. 113, lines -13, -12: replace "a-orbit" by "orbit".

p. 118, line 8 of 14.2.2: replace pb € Z by pb e Q.

p. 124, line 2 of 14.4.11: replace by e f ' X ) + 1 .

p. 128: Ref. 6, add: I.H.E.S. before 67.

p. 132, line 1 of 16.1.2(b): replace t r^*" 1 ) / 2 by t ^ " 1 ) / 2 .

p. 138 line -7, -5, -3: replace last C{M)) by C{M).

p. 138 line -2: replace all C{M)) by C{M) (three times).

p. 164, line 6 of 19.1.1, replace Sp\ by piS.

p. 171, line 4: replace Let be Aa by Let b e AA, b / 0.

p. 177 line -2: replace w(u)x{ji) (following the second =) by u){u) • x(v)-

p. 181 in lines 2, 5, 6: replace df,}9itb',n by

p. 181 in lines 2, 6: replace \b'\ by \(5'\.

p. 185 in line -8 , -9 : replace AI, AI', A'2J A'2' and any t G U(Ai - A"), s G U(Ai - A"),

by the following: A'1? A", X'2, X'i in X, any z^ in Z [I] such that A; - A'/ = vu Ai - A7/ = v2 (in X) and any t G U(i/i), s € U(i/2),

p. 190 line -1: replace M\ <g> T by UM\ <g> T.

At the end of p. 191 (after "...from 23.3.6") add a new subsection:

23.3.11.

We give a more detailed proof of Proposition 3.5.4 based on results in this chapter. Let 7TA',A" be as in 23.1.1. Prom the definition we see that if u E U satisfies ,A2 (w) = 0 for any AI, A2 in X then u = 0.

Let a = G N[/], b = G N[/], X e X, J(\a,b) = E i + ^ + E ^ y U ^ - v ^ ) . Let « G Da,MJ(A, a, 6). By 3.5.3 it is enough to show that u = 0. Hence it is enough to show that 7TA1,A2(w) = 0 for any AI,A2 in X. If (i, A2) = ai — hi for all i G I we have 7r\lix2(J(X2,a,b)) C P(X2,a,b) (notation of 23.3.3). Hence it is enough to show that for any £ G X we have HJP(£, a, b) = 0 where the intersection is taken over all a, b such that (i, (} — a^ — bi for all i. Us-ing the isomorphism <j) in 23.2.5 we see that we may assume that our root datum is simply connected. We identify X+ with N[J] by A M A where A = ^ enough to show that for any £ G X we have n A ' , A € A : + ; C = A ' - A - P ( C j A', A) = 0. For A', A G X+ such that £ = A'-A we have an isomorphism U l c -> f ® f = UM\ <g> My (see 23.3.1(c)) which by 23.3.5 carries P(C> A'5 A) onto T\ <g> f + f (g> Ty where T\ (resp. Ty) is the kernel of the canonical homomorphism f = UM\ (resp. f = My —> Ay). It is then enough to show that nA,A'ex+;C=A'-ApA <8> f + f ® Ty) = 0.

Now assume that x belongs to the last intersection. We have x — EI/GF I/'GF' ^ ® & where F, F' are finite subsets of N[/]. We can find A G X+ such that A + C e X+, A) > tr(i/) + tr(i/), Ei<*> A + C) > t r (v) + tr(i/) for any v G F, z/ G F'. If x ^ 0 then for such A we have x^TA(8)f + f<S) 7x+c. This shows that x = 0. This proves 3.5.4.

p. 200 line -5: replace last A' by A

p. 220: part (d) of Theorem 27.3.2 and the first two lines of the proof are missing. Thus, at the bottom of p. 220 (after 27.3.2(c)) add:

(d) The natural homomorphism C fl —> C/v~xC is an isomorphism. The element b\ ob[ = J2b2,b'2 ^fei.&i,62,63® satisfies the requirements of (a). This shows the existence in (a). It is also clear that the elements

p. 228: the diamond in line 2 of 28.2.6 should be bigger (of the same size as the diamond in line 4 of 28.2.6).

p. 237 in line 2 of 29.5.1: replace: on U[> A] for some A e X+ by on some two-sided ideal of finite codimension of U.

p. 240 first line of 30.1.7: remove one dot after "Example".

p. 246 line 8: replace x 6 iu by x E r$u

p. 320 in line 5 of 40.1.2: replace l(w) = 1 by l(w) = 0.

p. 320 in line 6 of 40.1.2: replace l(w) > 1 by l(w) > 0.

p. 325 line -10: replace [7] by [6].