平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (

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1 1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i P j P j P i = 0, X i P j P j X i = 1 h 2π δ ij 1 (1)

2 . 1 L 1 = X 2 P 3 X 3 P 2, L 2 = X 3 P 1 X 1 P 3, L 3 = X 1 P 2 X 2 P 1 L 1, L 2, L 3 L 1 L 2 L 2 L 1 = 1 h 2π L 3, L 2 L 3 L 3 L 2 = 1 h 2π L 1, L 3 L 1 L 1 L 3 = 1 h 2π L 2, X = 2π ( L1 + ) 2π ( 1L 2, Y = L1 ) 4π 1L 2, H = h h h L 3 HX XH =2X, HY Y H = 2Y, (2) XY Y X =H,. 2 ( ) ( ) X =, Y =, H = ( X, Y, H Mat(n, n, C) (2) P P 1 XP, P 1 Y P, P 1 HP P 1 XP = diag(x (1),..., X (s) ), P 1 Y P = diag(y (1),..., Y (s) ), P 1 HP = diag(h (1),..., H (s) ), ) 1 C n ) C[x 1, x 2,... ] X i = x i, P i = 1 h 1 2π x i 2 U(sl 2 ) 2

3 (X (i), Y (i), H (i) ) (1 i s) 0 l 0 0 l X =, Y = 0 1 l l 0 l l 2 H = l+2 l X, Y, H Mat(l+1, l+1, C) C l+1 V l l/2 sl 2 Hamiltonian E n (n = 1, 2,... ) V 1 V l l l {0, 2,..., 2n 2} l/2 l/2 = 0, 1, 2,... s p... (sharp, principal,...) 1.3 Lie X 1,..., X n X i X j X j X i = n k=1 ak ij X k Lie a k ij X i X j X j X i = (X j X i X i X j ) a k ij + ak ji = 0 X i (X j X k X k X j ) (X j X k X k X j )X i + X j (X k X i X i X k ) (X k X i X i X k )X j + X k (X i X j X j X i ) (X i X j X j X i )X k = 0 3

4 X 1,..., X n n p=1 3 ( ) a p jk aq ip + ap ki aq jp + ap ij aq kp = 0 {ψ i, ψ i} i Z. 4 ψ i ψ j + ψ j ψ i = 0, ψ i ψ j + ψ j ψ i = 0 ψ i ψ j + ψ j ψ i = δ ij (3) vac Fermion Fock ψ i vac = 0 (i < 0), ψ i vac = 0 (i 0), Fock H n = i Z : ψ i ψ i+n : = i<0 ψ i+n ψ i + i 0 ψ i ψ i+n : H n (n Z) :. 5 H m H n H n H m = mδ m+n,0 affine Kac Moody Lie Fock 3 Jacobi 4 Cψ i Cψ i Clifford 5 i 1 X i = 1 i H i P i = 1 h 1 2π H i H 0 X i = X i H 0 H 0 P i = P i H 0 X i P i (1) 4

5 Virasoro Lie Fock. 6 Wess-Zumino-Witten C. N. Yang Yangian U(sl 2 ) [F], [GRS] [JM] 20 Hecke Lie Lie Hecke A Mat(m, n, C) P Mat(m, m, C) Q Mat(n, n, C) P 1 AQ = E E rr E ij (i, j) 1 0 r P, Q A r A rank A 6 Virasoro Lie L n (n Z) L m L n L n L m = (m n)l m+n + m3 m δ m+n,0 12 5

6 A 1 r A A A A A Mat(n, n, C) P Mat(n, n, C) P 1 AP = diag(j 1,..., J s ) J i λ 1 λ 1 J(k, λ) = Mat(k, k, C) λ 1 λ Jordan J 1,..., J s P A Jordan 2.2 f 1 (X) = 0,..., f r (X) = 0 f 1,..., f r f f(x) = O f 1 (X) = O,..., f r (X) = O X f(x) = O f(p 1 XP ) = O X Jordan 2. (X 2I) 2 (X + 3I) = O X Jordan λ = 3, 2 λ = 3 X 2I X = 3I Jordan 1 ( 3) λ = 2 X + 3I (X 2I) 2 = O Jordan (2) ( ) J = ( ) { X = P 1 diag( 3,, 3, 2,, 2, J,, J)P P } 6

7 2.3 a(b + c) = ab + ac (a + b)c = ac + bc f(x) n A = C1 CX n 1 a(x) A b(x) A a(x)b(x) (mod f(x)) A A A C[X]/(f) f(x) = O X C[X]/(f) ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ). 7 C X 1,..., X n /(f 1,..., f r ) X i X j = n k=1 ak ij X k Lie a k ij (X ix j )X k = X i (X j X k ) X 1,..., X n n a p ij aq pk = p=1 n a p jk aq ip I = c i X i p=1 IX i = X i I = X i (1 i n) X 1,..., X n X i X j = n k=1 ak ij X k A = n i=1 CX i X i X j = n k=1 ak ij X k A = C X 1,..., X n / (Xi X j n a k ijx k 1 i, j n ) k=1 7C X 1,..., X n n 7

8 . 8 A A A E 1 = E 11, F = E 12, E 2 = E 22 E i E j = δ ij E i, E 1 +E 2 = I, E i F = δ i1 F, F E j = δ 2j F, F 2 = O (4) A (4) (E 1, E 2, F ) X 2 = O 2.4 Jordan X 2 = O Jordan X = P diag(0,, 0, J,, J)P 1 J = ( ) P X Mat(n, n, C) V C n V = Ker X = {v Xv = 0} X 2 = O Im X V V {e 1,..., e l } e l+1,..., e n {e 1,..., e n } C n e 1,..., e n T = (e 1,..., e n ) {e 1,..., e n } C n T XT 1 i l V Xe i = 0 l + 1 i n Xe i V a ij Xe i = a 11 e a l1 e l A Mat(l, n l, C) A = (a ij ) XT = (Xe 1,..., Xe n ) = (e 1,..., e n ) ( O O ) A = T O ( O O ) A T 1 XT = ( O A O O) T = ( P O O Q) ( ) (T T ) 1 X(T T O P 1 AQ ) = O O 8 B B V =CX 1 CX n X i V V 1 V V = B B A O 8

9 Jordan ( ) O E E rr X = P O O P 1 r = dim(im X) = n dim(ker X) = n l l Jordan Kostant X Mat(n, n, C) X n = O X Kostant 3. X H, Y (2) 1 X 2 = O Jordan X 2 = O X (X, Y, H) Lie 2.6 T (2, C) X 2 = O T (2, C) T (2, C) (4) E i E j = δ ij E i, E 1 +E 2 = I, E i F = δ i1 F, F E j = δ 2j F, F 2 = O (E 1, E 2, F ) X 2 = O 2.4 {e 1,..., e n } E 1, E 2 Mat(n, n, C) E 1 : n c i e i i=1 l c i e i, E 2 : i=1 n c i e i i=1 n i=l+1 c i e i, T 1 E 1 T = ( ) I O O O, T 1 E 2 T = ( ) O O O I F = X (E 1, E 2, F ) (4) T (2, C). 10 X U(sl 2 ) X T (2, C) T (2, C) 9 X 2 = O Im X Ker X n l l 10 rank X = n l F : Im E 2 Im E 1 9

10 4. (E 1, E 2, F ) (4) V = Im E 1 W = Im E 2 (1) V = Ker E 2 W = Ker E 1 C n = V W (2) F W V V {e 1,..., e l } W {f 1,..., f n l } V F e i = 0 (1 i l) W F (f 1,..., f n l ) = (e 1,..., e l )A A = E E rr. 11 V W F ( ) O E E rr F (e 1,..., e l, f 1,..., f n l ) = (e 1,..., e l, f 1,..., f n l ) O O F W 12 Ker F = V. 13 T (2, C) C Id C, C 0, 0 C T (2, C) X 2 = O T (2, C) Lie ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 11C n 8 F Ce i >< >: Ce i Cf i (1 i r) F f i = e i, F e i = 0, Cf i F Cf i (r < i n l) F f i = 0, Cf i 0 Ce i (r < i l) F e i = 0 0 Ce F i 12 A 13 Ker F V 14 T (2,C) 10

11 X = (X 1,..., X n ) 5. P P 1 X i P P ( ) P 1 X (1) i O X i P = O X (2) (1 i n) i 6. P P 1 X i P P ( ) P 1 X (1) i Y i X i P = O X (2) (1 i n) i ( ) ( ) ( ) E 1 = , E 2 =, F = T (2, C) F 1 T (2, C) E 1 = (1), E 2 = (0), F = (0), E 1 = (0), E 2 = (1), F = (0), 7. ( ) C X 1,..., X n /(f 1,..., f r ) A A Ind(A) Ind(A) A 11

12 3.2 X 2 = O T (2, C) X 2 = O Jordan J = ( ) Jordan X = P diag(0,, 0, J,, J)P 1 P = (e 1,..., e n ) J r V = Ker X V = ( n 2r i=1 V, ) r 1 Ce i j=0 Ce n 1 2j {e 1,..., e n 2r, e n 2r+1, e n 2r+3,..., e n 1 } {e n 2r+2,..., e n } C n e n 2j (0 j r 1) Xe n 2j = e n 1 2j e 2i+n 2r 1 (1 i r) f i = e i r (r < i n r) e 2i n (n r < i n) 1 i n r Xf i = 0 n r < i n i = n r+j (1 j r) Xf n r+j = Xe n 2(r j) = e n 1 2(r j) = f j Q = (f 1,..., f n ) Q 1 XQ Jordan Jordan X 2 = O T (2, C) C Id C, C 0, 0 C X W (0 C) ( ) (n 2r) C Id r C 12

13 Jordan P 1 XP = diag(0,..., 0, J,..., J) X i p F p G = x x p = 1 X p = I X Mat(n, n, F) Jordan p 0 J i (1 i p) 1 i Jordan J i = J(i, 1) i 1 J i = I + N i, N i = k=1 E k,k+1 1 i p N p i = O 17 J p i = I + p k=1 ( ) p Ni k = I + N p i = I k X = P diag(j 1,..., J 2,......, J p )P 1 Ind(FG) p G = x, y x p = y p = 1, xy = yx p = 2 16 T (2,C) 4 {e 1,..., e l, f 1,..., f n l } 8 >< Id F C Ce i Cf i (1 i r) Cf i Ce i F C 0 Cf i (r < i n l) Cf i 0, >: 0 C Ce i (r < i l) 0 Ce F i e i, f i e i, f i i p X 17 i > p N p i = E k,k+p O k=1 13

14 X = (1), Y = (1), X = , Y = n+1 (n 1) ( ) ( ) I n (I n 0) I n (0 I n ) X =, Y = O I n+1 O I n+1 ( (I In0 )) ( ( n+1 I 0 )) n+1 I X =, Y = n O O I n F { } 2n (n 1) ( ) ( ) I n I n I n J(n, λ) X =, Y = (λ F) O I n O I n ( ) ( ) I n J(n, 0) I n I n X =, Y = O O I n C p p Ind(FG) 8. f 1,..., f r A = F X 1,..., X n /(f 1,..., f r ) Ind(A) Ind(FG) A n X n+1 = O n+1 = 2N, n n = 2N A M M (i) = F N M = M N (i = 1, 3), M (2) = F N+1 M = M N+1 FG F (M) F (M) = M (1) M (2) M (3) F (M) X Y X p = I, Y p = I, XY = Y X I n I n 14

15 M (1) 0 m 1 m 1 m 1 m (X 1) = 1 M (2), (Y 1) = M (2) m N m N m N 0 M (2) (X 1) (Y 1) m N m 1 m N+1 m 1 m N+1 M (3) (X 1) m 1 m N m 1 + X 1 m N+1 m = 2 + X 2 m N+1 M (3) m N + X N m N+1 X N+1 m 1 + m 2 X N+2 m 1 + m 3 = M (3) X 2N m 1 + m N+1 0 = 0, (Y 1) m 1 m N = 0 9. FG, A M, N A (1) M F (M). 18 (2) F (M) F (N) F (M) F (N) X Y M N M N X 1,..., X n Ind A Ind(FG) S n F (M) EndFG(F (M)) 19 {1, 2,..., n} S n 15

16 10. f 1,..., f r Ind (F X 1,..., X n /(f 1,..., f r )) Ind(FS n ) n 2p (p 3) n 6 (p = 2) 3.4 Hecke Hecke S n S n σ i (1 i < n) i i+1 σ i : i+1 i k k (k i, i+1) σ 2 i = 1, σ i σ j σ i = σ j σ i σ j (j = i ± 1) σ i σ j = σ j σ i (j i ± 1) FS n = F σ 1,..., σ n 1 /I I I = ( σi 2 1, σ i σ i±1 σ i σ i±1 σ i σ i±1, σ i σ j σ j σ i (j i ± 1) ). q C \ {0} Hecke H n (q) H n (q) = C σ 1,..., σ n 1 /I q I q I q = ( σi 2 (q 1)σ i q, σ i σ i±1 σ i σ i±1 σ i σ i±1, σ i σ j σ j σ i (j i ± 1) ) 16

17 11. q 1 q n = 1 n e. 20 f 1,..., f r Ind (C X 1,..., X n /(f 1,..., f r )) Ind(H n (q)) n 2e (e 3) n 6 (e = 2) e FS n F e H n (q) A Hecke Lie Lie Kazhdan-Lusztig Brylinski-Kashiwara, Beilinson-Bernstein Jones Temperly-Lieb H n (q) P 1 (C) Wess-Zumino-Witten n Knizhnik-Zamolodchikov H n (q) Drinfeld-Kohno Lie Lie Lie O Auslander-Reiten translation quiver. 21 community 20 n e = 21 Gabriel 17

18 Hall Hall 4 Weil Deligne. 23 Lie 24 T (2, C) E i E j = δ ij E i, E 1 +E 2 = I, E i F = δ i1 F, F E j = δ 2j F, F 2 = O V 1 : E 1 = (1), E 2 = (0), F = (0), V 2 : E 1 = (0), E 2 = (1), F = (0), ( ) ( ) ( ) V 12 : E 1 =, E 2 =, F = Ind(T (2, C)) = {V 1, V 2, V 12 } q F q q Ind(T (2, F q )) = {V 1, V 2, V 12 }. 25 Hall Hall A = F q X 1,..., X n /(f 1,..., f r ) A V 1, V 2, V 3 X i U U (1 i n) U V 3 (a) U V 1 (b) V 3 /U V 2 22 Hall 23 Hecke Weil 24 Lie F q 18

19 h V3 V 1,V 2 (q) H V3 V 1,V 2 (x) q H V3 V 1,V 2 (q) = h V3 V 1,V 2 (q) Hall A V [V ] [V 1 ] [V 2 ] = V 3 H V3 V 1,V 2 (1)[V 3 ] A Hall T (2, F q ) V 12 E 1, E 2, F F 2 q ( ) ( ) ( ) E 1 =, E 2 =, F = ( 1 ) 0 E1, E 2, F 1, 0, 0 ( 1 0) V1 V 12 ( U = F 1 q 0) V12 /U E 1 = (0), E 2 = (1), F = (0) V 12 /U V 2 ( ) ( ) ( ) ( ) ( ) ( ) a 0 a a a 0 E 1 =, E 2 =, F = b 0 b b b 0 ( a b) ( a b) = ( 0 0) V2 V 12 h V 12 V 1,V 2 (q) = 1, h V 12 V 2,V 1 (q) = 0, h V1 V2 V 1,V 2 (q) = 1, h V1 V2 V 2,V 1 (q) = 1, h V 1 V 1 V 1,V 2 (q) = 0, h V 1 V 1 V 2,V 1 (q) = 0, h V 2 V 2 V 1,V 2 (q) = 0, h V 2 V 2 V 2,V 1 (q) = 0, [V 1 ] [V 2 ] = [V 12 ] + [V 1 V 2 ], [V 2 ] [V 1 ] = [V 1 V 2 ] 26 U V 3 /U 19

20 [V 1 ] [V 2 ] [V 2 ] [V 1 ] = [V 12 ] h V 12 V 1,V 1 (q) = 0, h V 12 V 2,V 2 (q) = 0, h V1 V2 V 1,V 1 (q) = 0, h V1 V2 V 2,V 2 (q) = 0, h V 1 V 1 V 1,V 1 (q) = q+1, h V 1 V 1 V 2,V 2 (q) = 0, h V 2 V 2 V 1,V 1 (q) = 0, h V 2 V 2 (q) = q+1, V 2,V 2 [V 1 ] [V 1 ] = 2[V 1 V 1 ], [V 2 ] [V 2 ] = 2[V 2 V 2 ] Hall C x 1, x 2 / ( x 2 1x 2 2x 1 x 2 x 1 + x 2 x 2 1, x 2 2x 1 2x 2 x 1 x 2 + x 1 x 2 ) 2, 27 E 12, E 23, E 13 E 12 [V 1 ], E 23 [V 2 ], E 13 [V 12 ] Hall. 28 E 12, E 23, E 13 T (3, C) E ij E kl = δ jk E il E 12 E 23 E 23 E 12 = E 13, E 12 E 13 E 13 E 12 = 0, E 23 E 13 E 13 E 23 = 0 Lie sl 3 C X, Y, Z / (XY Y X Z, XZ ZX, Y Z ZY ) = C X, Y / ( X 2 Y 2XY X + Y X 2, Y 2 X 2Y XY + XY 2) 13. T (2, F q ) Hall U(sl 3 ). 29 C X, Y, Z / (XY Y X Z, XZ ZX, Y Z ZY ) sl 3 Lie Lie U(sl 3 ) 27 x i [V i ] (i = 1, 2) x 1 x 2 x 2 x 1 [V 12 ]. 28 E 2 12 E 23 2E 12 E 23 E 12 +E 23 E 2 12 = O, E2 23 E 12 2E 23 E 12 E 23 +E 12 E 2 23 = O. 29 U + (sl 3 ) 20

21 4.3 Gabriel Mat(n, n, C) T (n, C) E 11, E 22,..., E nn E 12, E 23,..., E n 1,n T (n, C) = E 1,..., E n, F 1,..., F n 1 /I I = (E i E j δ ij E i, E i F j δ ij F j, F j E i δ i,j+1 F j, F j F k (j+1 k)) V i = Im E i i j +1 F j (V i ) = 0 i = j+1 F j (V j+1 ) V j 2.6 T (n, C) V n F Vn 1 F F V1 14. T (n, C) 0 0 C Id Id C 0 0 ( ) n+1 2 T (n, C) T (n, F q ) Hall V ij (n+1 i > j 1) F q (i > k j) Im E k = 0 (k i, k < j) Hall V = i>j V n ij ij (n ij Z 0 ) Hall sl n+1 Lie U + (sl n+1 ) = C X 1,..., X n /I I = ( Xi 2 X i±1 2X i X i±1 X i + X i±1 Xi 2, X i X j X j X i (j i ± 1) ) Dynkin Gabriel 21

22 Lie Lie Lie Gabriel [ ] 4.4 Lie. 30 Lie Lie [ ] 2002 [A] Susumu Ariki, Representations of Quantum Algebras and Combinatorics of Young Tableaux, AMS University Lecture Series, [ARS] M. Auslander, I. Reiten and S. Smalo, Representation Theory of Artin Algebras, Cambridge studies in Advanced Mathematics, [F] J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge Monographs on Mathematical Physics, [GRS] C. Gómez, M. Ruiz-Altaba and G. Sierra, Quantum Groups in Two-dimensional Physics, Cambridge Monographs on Mathematical Physics, [H] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, [JM] M. Jimbo and T. Miwa, Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Mathematics, S 2 Riemann SO(3)/SO(2) Laplace Legendre 22

23 [K] V. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press,

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