II Lie Lie Lie ( ) 1. Lie Lie Lie

Transcription

1 II Lie

2 II Lie Lie Lie ( ) 1. Lie Lie Lie

3 i Lie Lie 4 3 Lie Lie Lie Lie Lie Lie 35

4 1 1 C Lie Lie 1.1 Hausdorff M M {(U α, φ α )} α A (1) M = α A U α, (2) α A φ α U α R n φ α (U α ) R n φ α U α φ α (U α ) (3) α, β A U α U β φ β φ 1 α φ β (U α U β ) : φ α (U α U β ) (M, {(U α, φ α )} α A ) n {(U α, φ α )} α A M M dim M {(U α, φ α )} α A M 1.2 (M, {(U α, φ α )} α A ), (N, {(V i, ψ i )} i I ) M N (M N, {(U α V i, φ α ψ i )} (α,i) A I ) M N 1.3 (M, {(U α, φ α )} α A ) n M V ψ : V R n (1) ψ(v ) R n ψ : V ψ(v ) (2) α A ψ φ 1 α : φ α (U α V ) ψ(u α V ) (V, ψ) M ψ R n ψ 1,..., ψ n V (V ; ψ 1,..., ψ n ) 1.4 m M n N f (1) f (2) f(u) V M (U, φ) N (V, ψ) R m φ(u) R n ψ(v ) ψ f φ 1 C

5 2 1. f C M N C f f 1 : N M f 1 C f M N 1.5 M x C F x R v f, g F x r R v(f + g) = v(f) + v(g), v(rf) = rv(f), v(fg) = v(f)g(x) + f(x)v(g) v M x T x (M) M x 1.6 n M x (U, φ) U x F x f f φ 1 φ(v ) C (x 1,..., x n ) R n v i (f) = (f φ 1 ) x i v i T x (M) v i x i x 1.7 n M x T x (M) n (U; x 1,..., x n ) M x x i x (1 i n) T x (M) 1.8 F M N C x M, y = F (x) N v T x (M) φ(x) v (f) = v(f F ) (f F F (x) ) v : F F (x) R v N F (x) v = df x (v) df x : T x (M) T F (x) (N) 1.9 M O x M x X x X X O M (U; x 1,..., x n ) x X x (x i ) i U O C

6 X M x M (U; x 1,..., x n ) 1.7 X x T x (M) n X x = a i (x) x i x i x (1 i n) X x C n X x (x j ) = a i (x) x i x j = a j (x) x U x i x (1 i n) C 1.11 M M C C (M) M X(M) (1) M X, Y f C (M) x Z(f) = X(Y (f)) Y (X(f)) Z M Z = [X, Y ] [X, Y ] X Y Lie (2) (1) [, ] : X(M) X(M) X(M) X, Y, Z X(M) [X, Y ] + [Y, X] = 0, [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 Jacobi (1) Z X Y X = i a i, x i Y = j b j x j Z(f) = i = i,j = i,j a i x i ( j ) f b j x j j ) 2 f ( bj f a i + b j x i x j x i x j ( ) b i a i f a j b j x j x j x i ( ) f b j a i x j x i i ( ) ai f 2 f b j + a i x i,j j x i x j x i

7 4 2. Lie Lie Z Lie [ a i, ] b j = ( ) b i a i a j b j. x i i x j j x i,j j x j x i 2 Lie Lie 2.1 G G G G; (x, y) xy, G G; x x 1 C G Lie ( e ) 2.2 V V GL(V ) Lie GL(R n ) GL(n, R) GL(V ) GL(V ) dim V = n End(V ) = {f : V V f } V End(V ) End(V ) n M n (R) M n (R) R n2 End(V ) R n2 End(V ) R n2 End(V ) End(V ) det : End(V ) R End(V ) GL(V ) = {f End(V ) det f 0} GL(V ) End(V ) GL(V ) n 2 GL(V ) GL(V ) GL(V ); (x, y) xy End(V ) C GL(V ) GL(V ); x x 1 End(V ) C (Cramer ) 2.3 Lie G g L g, R g L g : G G; x gx, R g : G G; x xg

8 5 g G X G g (dl g ) x (X x ) = X gx (x G) (dr g ) x (X x ) = X xg (x G) 2.4 Lie G e (U; x 1,..., x n ) g G (L g (U); x 1 L 1 g,..., x n L 1 g ) g 2.5 g [, ] : g g g X, Y, Z g [X, Y ] = [Y, X], [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0 g Lie Lie g h [, ] h g Lie Lie g Lie g Lie 2.6 M X(M) Lie [, ] Lie 2.7 V End(V ) X, Y [X, Y ] = XY Y X End(V ) Lie Lie gl(v ) gl(r n ) gl(n, R) [, ] : End(V ) End(V ) End(V ) End(V ) X, Y, Z [X, Y ] = XY Y X = [Y, X], [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = [XY Y X, Z] + [Y Z ZY, X] + [ZX XZ, Y ] = XY Z Y XZ ZXY + ZY X + Y ZX ZY X XY Z + XZY = 0. +ZXY XZY Y ZX + Y XZ End(V ) [, ] Lie

9 6 2. Lie Lie 2.8 G Lie G g g G Lie X(G) Lie α : g T e (G); X X e dim g = dim T e (G) = dim G 2.9 M, N f : M N M X, Y N X, Ỹ df x (X x ) = X f(x), df x (Y x ) = Ỹf(x) (x M) df x ([X, Y ] x ) = [ X, Ỹ ] f(x) (x M). (V ; y 1,..., y n ) N U = f 1 (V ) x i = y i f (1 i n) (U; x 1,..., x n ) M X, Y U n n X = a i, Y = b i x i x i V ( n ) X = df(x) = df a i = x i Ỹ = n (a i f 1 ) y k = x i y k n (b j f 1 ). y j j=1 n (a i f 1 ) y i Lie [ X, Ỹ ] = ( (a j f 1 ) (b i f 1 ) (b j f 1 ) (a ) i f 1 ). y i,j j y j y i ( ( ) ) b i a i df([x, Y ]) = df a j b j x i,j j x j x i = ( (a i f 1 ) b i f 1 (b j f 1 ) a ) i f 1 x i,j j x j y i

10 7 b i x j f 1 = b i f 1 x j, df([x, Y ]) = [ X, Ỹ ] a i x j f 1 = a i f 1 x j 2.8 g X(G) X, Y g g G (dl g ) x (X x ) = X gx (dl g ) x (Y x ) = Y gx (x G) L g : G G 2.9 (dl g ) x ([X, Y ] x ) = [X, Y ] gx (x G) [X, Y ] g g X(G) Lie α X g, α(x) = 0 g G X g = (dl g ) e (X e ) = 0 X = 0 Kerα = 0 α X T e (G) X g = (dl g ) e (X) X α (U; x 1,..., x n ) G e G G G; (x, y) xy e V V V = {xy x, y V } U X = n ξ i x i 2.4 g G (L g (V ); x 1 L 1 g,..., x n L 1 g ) g y i = x i L 1 g gx L g (V ) (x V ) X gx = (dl gx ) e (X) = d(l g L x ) e (X) = (dl g ) x (dl x ) e (X) ( n = (dl g ) x ξ i (x ) j L x ) (e) x i,j=1 i x j x n = ξ i (x j L x ) (e)(dl g ) x x i x j. x i,j=1 ( (dl g ) x ) ( ) x j (y k ) = x x j (y k L g ) = δ jk x e

11 8 3. Lie (dl g ) x x j = x y j gx X gx = n i,j=1 ξ i (x j L x ) (e) x i y j. gx G G G; (x, y) xy = L x (y) C i, j V R; x (x j L x ) (e) x i C j L g (V ) R; gx n ξ i (x j L x ) (e) x i C X G X α 2.10 Lie G Lie g Lie G Lie 2.11 Lie C f : G H f Lie f f 1 f 1 Lie f Lie Lie G H Lie f : g h [f(x), f(y )] = f([x, Y ]) (X, Y g) f Lie f f 1 f Lie Lie g h 3 Lie 3.1 G Lie U G e G = {U n n N} U n = {g 1... g n g i U} U 1 = {g 1 g U} e V = U U 1 e H = {V n n N} V U G = H g, h H m, n g V m, h V n gh V m+n H g = g 1... g m, g i V g 1 = gm 1... g1 1 g 1 i V

12 9 g 1 V m H H G V e V H h H L h (V ) h L h (V ) H H G G H G = H ( {L g (H) g G, g / H}) L g (H) G H G G G = H G Lie G G 0 G 0 G Lie τ : G G G; (g, h) gh 1 τ G 0 G 0 G G (e, e) τ(g 0 G 0 ) e = τ(e, e) τ(g 0 G 0 ) τ(g 0 G 0 ) G 0 G 0 G g G L g R g 1 L g R g 1(G 0 ) e = L g R g 1(e) L g R g 1(G 0 ) L g R g 1(G 0 ) G 0 G 0 G G 0 G τ : G G G C G 0 τ G0 : G 0 G 0 G 0 C G 0 Lie G 0 G Lie Lie G G V n Lie GL(V ) GL(n, R) Lie gl(v ) gl(n, R) v 1,..., v n V f End(V ) f v 1,..., v n R(f) f[v 1,..., v n ] = [v 1,..., v n ]R(f) R : End(V ) End(R n ) R R Lie R : gl(v ) gl(n, R)

13 10 4. R GL(V ) Lie R GL(V ) : GL(V ) GL(n, R) 4.2 GL(n, R) 2.2 gl(n, R) T e (GL(n, R)) gl(n, R) Lie GL(n, R) Lie g X gl(n, R) X g X g = (dl g ) e (X) (g GL(n, R)) : gl(n, R) g ; X X Lie 2.8 GL(n, R) = α 1 Lie (i, j)- 1 0 n E ij {E ij 1 i, j n} gl(n, R) {x ij 1 i, j n} x ij gl(n, R) GL(n, R) gl(n, R) e GL(n, R) X gl(n, R) t R e + tx GL(n, R) g GL(n, R) ( ) d X g = (dl g ) e (X) = (dl g ) e dt t=0 (e + tx) = d dt L g(e + tx) = d t=0 dt t=0 (g + tgx) n n n = x ij (gx) = x ik (g)x kj (X) g i,j=1 x ij i,j=1 k= Lie X, Y gl(n, R), g GL(n, R) ( n ) [ X, n n x ik (g)x kj (Y ) Ỹ ] k=1 g = x pr (g)x rq (X) x i,j,p,q=1 r=1 pq ( n ) n x ik (g)x kj (X) k=1 x pr (g)x rq (Y ) x r=1 pq x ij g { n n } n = x ir (g)x rk (X)x kj (Y ) x ir (g)x rk (Y )x kj (X) i,j=1 k,r=1 k,r=1 x ij. g x ij g

14 11 = n x ij (gxy gy X) i,j=1 = [X, Y ] g. x ij = g n x ij (g[x, Y ]) i,j=1 x ij g [ X, Ỹ ] = [X, Y ] Lie Lie : gl(n, R) g Lie gl(n, R) Lie GL(n, R) Lie g gl(n, R) GL(n, R) Lie 4.1 V gl(v ) GL(V ) Lie M X M I M c : I M dc dt (t) = X c(t) (t I) c X 5.2 M X M t 0 M x M X c : I M t 0 I, c(t 0 ) = x c 1, c 2 : I M c 1 (t 0 ) = c 2 (t 0 ) = x X c 1 = c 2 x M (U; x 1,..., x n ) X U X x = n a i (x) x i (x U) x U dc dt (t) = X c(t) (t I) n d(x i c(t)) dt x i = c(t) n a i (c(t)) x i c(t)

15 12 5. c d(x i c(t)) dt = a i (c(t)), x i c(t 0 ) = x i (x) (1 i n) Euclid a i C t 0 I c : I U c M (U; x 1,..., x n ) c 1 (I), c 2 (I) U x 1,..., x n dc 1 dt (t) = X c 1 (t), dc 2 dt (t) = X c 2 (t) (t I) Euclid c 1 = c 2 c 1 (I), c 2 (I) M 1 t 0 < s, s I 0 < ε t 0 < t 1 < < t k = s I i = (t i 1 ε, t i + ε) (1 i k) i c 1 (I i ), c 2 (I i ) M 1 c 1 (t 1 ) = c 2 (t 1 ),..., c 1 (t k ) = c 2 (t k ) c 1 (s) = c 2 (s) t 0 > s, s I c 1 (s) = c 2 (s) c 1 = c R Lie R Lie G Lie G 5.4 G Lie Lie g Lie g G 1 1 X g X c : R G c(0) = e 1 c G X g c G c 2.8 dc (0) g X c dt (1),(2) (1) X g X c : R G c(0) = e c G (2) 2 (1) 5.2 δ > 0 X a : ( δ, δ) G a(0) = e s < δ/2 s 1 b 1 (t) = a(s + t), b 2 (t) = a(s)a(t) ( t < δ/2)

16 13 t b 1 (t) X a(t) X L g a(t) X ( ) d d dt L ga(t) = (dl g ) a(t) dt a(t) = (dl g ) a(t) (X a(t) ) = X Lg a(t). b 2 (t) X 5.2 b 1 (0) = a(s) = a(s)e = a(s)a(0) = b 2 (0) b 1 (t) = b 2 (t) ( t < δ/2). a(s + t) = a(s)a(t) = a(t)a(s) ( s, t < δ/2) t R t/k < δ/2 k ( ) k t c(t) = a k c : R G t R t/k < δ/2, t/l < δ/2 k, l ( ) k ( ) kl ( ) l t t t a = a = a k kl l c k c(0) = a(0) = e c G t R t/k < δ/2 k t I s I s/k < δ/2 s I c(s) = a(s/k) k c I C c : R G C s, t R s/k, t/k < δ/4 k s/k, t/k, (s + t)/k < δ/2 ( s ) ( ) k k ( t ( s ) ( )) k t c(s)c(t) = a a = a a k k k k ( ) k s + t = a = c(s + t). k c : R G G c X s R t (s δ, s + δ) c(t) = c(s)c(t s) = L c(s) (c(t s)) ( ) d dt c(t) d = (dl c(s) ) e t=s dt t=0 c(t) = (dl c(s) ) e (X e ) = X c(s).

17 14 6. c X (2) X g G c dc(0) = X dt e c g X G c g X X e = dc(0) dt (1) g X G c dc(0) = X dt e c X X G c n M n (C) X M n (C) X = max{ Xv v C n, v 1} M n (C) e X = k=0 1 k! Xk e X 6.2 X, Y M n (C) P M n (C) (1) e P XP 1 = P e X P 1. (2) XY = Y X e X+Y = e X e Y (3) e X (e X ) 1 = e X. (4) d dt etx = e tx X = Xe tx. (1) M n (C) ( e P XP 1 ) 1 = k! (P XP 1 ) k 1 = k! P Xk P 1 1 = P k! Xk k=0 = P e X P 1. (2) XY = Y X (X + Y ) k = i=0 k=0 k=0 P 1 k ( ) k X i Y k i = k! i i!j! Xi Y j = k! 1 1 i! Xi j! Y j i+j=k i+j=k

18 15 ( e X+Y 1 = k! (X + Y 1 )k = 1 ) ( i! Xi j! Y ) j 1 1 = i! Xi j! Y j k=0 = e X e Y. k=0 i+j=k (3) X( X) = X 2 = ( X)X (2) e = e 0 = e X X = e X e X (e X ) 1 = e X (4) e tx i=0 j=0 d dt etx = d dt k=0 1 k! tk X k = = e tx X = Xe tx. 1 d k! dt tk X k = k=0 k=1 1 (k 1)! tk 1 X k = k=0 1 k! tk X k GL(n, R) GL(n, R) gl(n, R) X gl(n, R) = T e (GL(n, R)) GL(n, R) X 2.8 X g = gx (g GL(n, R)) X GL(n, R) c dc(t) dt = c(t)x (t R), c(0) = e c(t) = e tx 6.4 n A γ A (t) γ A (t) γ A (t) = (t λ 1 ) p 1... (t λ k ) p k : 1 γ A (t) = h 1(t) (t λ 1 ) + + h k(t) p 1 (t λ k ) p k h i (t) p i 1 γ A (t) 1 = h 1 (t) (t λ 1 ) + + h γ A (t) k(t) p 1 (t λ k ) p k γ A (t) (t λ i ) p i γ A(t) (t λ i ) t p i h i (t) h i (t) h i (t) t = λ i

19 16 7. t = λ i p i 1 γ A (t) π i (t) = h i (t) π (t λ i ) p i (t) t i 1 = π 1 (t) + + π k (t), (t λ i ) p i π i (t) = h i (t)γ A (t) P i = π i (A) I n = P P k. Cayley-Hamilton (A λ i I n ) p i P i = h i (A)γ A (A) = 0. e ta = = = k e ta P i = k e tλ ii n +t(a λ i I n ) P i k e λ iti n e t(a λ ii n ) P i = k k e λ it p i 1 e λ t it j j! (A λ ii n ) j P i. j=0 j=0 t j j! (A λ ii n ) j P i G Lie Lie g X g 5.4 X c : R G c(0) = e exp X = c(1) exp : g G exp Lie G GL(n, R) Lie gl(n, R) X e tx GL(n, R) 7.3 G Lie Lie g X g 5.4 G t exp tx X g 5.4 G t c(t, X) s R d dt c(st, X) = sx c(st,x) (t R)

20 17 t c(st, X) sx g c(st, X) = c(t, sx) t = 1 c(s, X) = c(1, sx) = exp sx. X g G t exp tx 7.4 G Lie Lie g G exp : g G C X 1,..., X n g u 1,..., u n u 1,..., u n g G (U; x 1,..., x n ) e U, x 1 (e) = = x n (e) = 0 n u i X i g c(t; u 1,..., u n ) d dt c(t; u 1,..., u n ) = n u i (X i ) c(t;u1,...,u n ) X i U n (X i ) x = a ji (x) j=1 x j x c(t; u 1,..., u n ) d dt x j(c(t; u 1,..., u n )) = n u i a ji (c(t; u 1,..., u n )) (t, u 1,..., u n ) (x 1 (c(t; u 1,..., u n )),..., x n (c(t; u 1,..., u n ))) 0 C ( n ) exp u i X i = c(1; u 1,..., u n ) exp : g G 0 N C X g p 1 p X N X O 1 p O N ( ) p 1 exp(z) = exp p Z (Z O) exp O C exp : g G C

21 Lie G Lie g G exp g 0 G e X g = T 0 (g) d exp 0 (X) = d dt exp(tx) t=0 = X e = α(x) d exp e = α : g T e (G) 2.8 d exp e exp g 0 G e 7.6 G Lie Lie g 7.5 exp g 0 G e ( U g n ) X 1,..., X n exp u i X i (u 1,..., u n ) U u 1 (e) = = u n (e) = 0 (u 1,..., u n ) g X 1,..., X n G (U; u 1,..., u n ) G 7.7 G Lie Lie g X, Y g, f C (G), g G (Xf)(g) = d dt t=0 f(g exp tx) ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) s=t=0 X g t g exp tx t = 0 (Xf)(g) = X g (f) = d dt t=0 f(g exp tx) 2 (XY f)(g) = d ds s=0 (Y f)(g exp sx) = f(g exp sx exp ty ) s t = ( ) t s f(g exp sx exp ty (exp sx) 1 exp sx) s=0 t=0 = ( t s f(g exp sx exp ty (exp sx) 1 ) s=0 s=t=0 + ) s s=0 f(g exp ty exp sx) (Leibniz ) t=0 = s t f(g exp sx exp ty (exp sx) 1 ) + (Y Xf)(g). s=t=0

22 19 ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ). s=t=0 7.8 Lie G G Lie g X, Y [X, Y ] = 0 f C (G), g G ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) = f(g exp ty ) s t = 0. s=t=0 s=t=0 [X, Y ] = Lie g X, Y [X, Y ] = 0 g 7.8 Lie Lie Lie Lie Lie Lie 8.2 G, H Lie Lie g, h f : G H Lie 2.8 α G : g T e (G), α H : h T e (H) df = α 1 H df e α G : g h Lie X g df e (X e ) T e (H) H df(x) g G x G df g (X g ) = df g (dl g ) e (X e ) = d(f L g ) e (X e ). (f L g )(x) = f(gx) = f(g)f(x) = (L f(g) f)(x) df g (X g ) = d(f L g ) e (X e ) = d(l f(g) f)(x e ) = (dl f(g) ) e df e (X e ) = df(x) f(g).

23 20 8. X, Y g df g (X g ) = df(x) f(g), df g (Y g ) = df(y ) f(g) (g G) Lie ( 2.9 ) df g ([X, Y ] g ) = [df(x), df(y )] f(g) (g G) df e ([X, Y ] e ) = [df(x), df(y )] e 2.8 [df(x), df(y )] H df([x, Y ]) = [df(x), df(y )]. df : g h Lie 8.3 Lie f : G H df = α 1 H df e α G : g h f 8.2 Lie Lie 8.4 A, B, C Lie Lie a, b, c A a f : A B, g : B C Lie d(g f) = dg df : a c d(id A ) = α 1 A d(g f) = α 1 C = α 1 C d(id A) e α A = α 1 A d(g f) e α A = α 1 C dg e α B α 1 B id T e (A) α A = id dg e df e α A df e α A = dg df 8.5 A, B Lie Lie a, b f : A B Lie df : a b Lie df d(f 1 ) = d(f f 1 ) = d(id B ) = id d(f 1 ) df = id d(f 1 ) = df 1 df Lie 8.6 G, H Lie Lie g, h f : G H Lie f(exp X) = exp(df(x)) (X g) exp G exp H

24 21 Lie Lie ( 8.1) t f(exp tx) H 5.4 Y h f(exp tx) = exp ty (t R) t = 0 ( ) = d dt t=0 f(exp tx) = df e (X e ) ( ) = d dt exp ty t=0 = Y e. df e (X e ) = Y e df(x) = Y f(exp tx) = exp(tdf(x)) t = 1 f(exp X) = exp(df(x)) 8.7 Lie G V G GL(V ) Lie G Lie g V g gl(v ) Lie g 8.8 Lie g X ad(x)(y ) = [X, Y ], Y g ad(x) gl(g) ad : g gl(g) Lie X, Y, Z g [ad(x), ad(y )](Z) = ad(x)ad(y )(Z) ad(y )ad(x)(z) = [X, [Y, Z]] [Y, [X, Z]] = [X, [Y, Z]] + [Y, [Z, X]] = [Z, [X, Y ]] (Jacobi ) = ad([x, Y ])(Z) [ad(x), ad(y )] = ad([x, Y ]) ad Lie 8.9 Lie g ad : g gl(g) g 8.10 Lie G g Ad(g) = d(l g R g 1) G Lie g Ad(g) GL(g) g exp(x)g 1 = exp(ad(g)x) (g G, X g) Ad : G GL(g) Lie Ad g L g R g 1 G 8.5 Ad(g) g Ad(g) GL(g) 8.6 g exp(x)g 1 = exp(ad(g)x) (g G, X g)

25 g, h G Ad(gh) = d(l gh R (gh) 1) = d(l g L h R g 1 R h 1) = d(l g R g 1 L h R h 1) = d(l g R g 1) d(l h R h 1) = Ad(g) Ad(h) Ad : G GL(g) Ad C GL(g) g X 1,..., X n θ 1,..., θ n GL(g) R; u θ i (u(x j )) G R; g θ i (ad(g)(x j )) C 2.8 θ i (Ad(g)(X j )) = θ i (α 1 G dl g dr g 1 α G (X j )) g C Ad g 7.7 X, Y g, f C (G), g G ([X, Y ]f)(g) = s t f(g exp sx exp ty (exp sx) 1 ) s=t=0 = ( ) f(g exp(ad(exp sx)ty )) s t t=0 s=0 = s ((Ad(exp sx)y )f(g)) s=0 = s ((Ad(exp sx)y )) s=0 f(g) d Ad(exp sx)y ds = [X, Y ] = ad(x)(y ) s=0 d Ad(exp sx) ds = ad(x) s=0 Ad ad 8.11 Lie G Ad : G GL(V ) G 8.12 V GL(V ) 7.2 GL(V ) g GL(V ), X gl(v ) Ad(g)X = d dt (getx g 1 ) = d t=0 dt t=0 etgxg 1 = gxg 1.

26 23 9 Lie 9.1 Lie H Lie G Lie H G H G 9.2 G Lie Lie g G Lie H ι : H G dι : h dι(h) Lie ι dι : h g 8.2 Lie H G dι : h dι(h) Lie dι(h) Lie H Lie dι h dι(h) 9.4 G Lie H G Lie G, H Lie g, h exp G, exp H X h exp G (X) = exp H (X) ι : H G ι Lie 8.6 X h ι(exp H (X)) = exp G (dι(x)) ι dι exp G (X) = exp H (X) 9.5 G Lie Lie g g = a + b g ϕ(x, Y ) = exp(x) exp(y ) C ϕ : a b G a, b 0 U, V G e W ϕ U V W dϕ 0 (X, Y ) a b dϕ 0 (X, Y ) = dϕ 0 ((X, 0) + (0, Y )) = dϕ 0 (X, 0) + dϕ 0 (0, Y ) = d dt exp(tx)) t=0 + d dt exp(ty ) t=0 = X e + Y e = α(x + Y ) dϕ 0 = α : g T e (G) 2.8 dϕ 0 a, b 0 U, V G e W ϕ U V W 9.6 G Lie H G H G H Lie

27 24 9. Lie g 1 { S = X g X n g {0}, exp X n H, lim X n = 0, lim n n } X n X n = X X S exp tx H (t R) X X n g {0}, exp X n H, lim X n = 0, lim n n n X n = X t R t = m n X n + r n, 0 r n < X n m n r n lim X n = 0 lim r n = 0 lim m n X n = t n n n exp X n tx = lim m n X n lim n n X n = lim m nx n. n exp tx = lim n exp m n X n = lim n (exp X n ) m n H (H ). h = {tx t R, X S} h g h X, Y h {0} exp tx, exp ty H (t R) t exp tx exp ty t = 0 e H 7.5 exp g 0 G e 0 I g c : I g c(0) = 0, exp(c(t)) = exp tx exp ty (t I) t = 0 ( ) dc α dt (0) = X e + Y e = α(x + Y ) 2.8 dc c(t) (0) = X + Y lim = X + Y dt t 0 t n 1 I n Z n = c( 1 ) n Z n 0 ( ( )) 1 exp Z n = exp c H, lim Z n = lim c n n lim n Z n Z n = lim n n ( ) 1 = c(0) = 0, n c ( ) ( 1 c 1 ( n c 1 ) = lim n) n 1 n n 1 n c ( 1 n ) = X + Y X + Y X+Y S X + Y h h g X+Y H H G g h h 1 : g = h + h 9.5 h, h 0 U, V G

28 25 e W ϕ : h h G; (X, Y ) exp(x) exp(y ) U V W h 0 N exp(n) H e exp(n) H e Euclid e e g n g n H, g n / exp(n) U N, g n W U g n X n U, Y n V g n = exp X n exp Y n g n / exp(n) Y n 0 Yn 1 Y n Y n X = lim n Y n Y n Y n h exp Y n = (exp X n ) 1 g n H, lim Y n = 0 X S h n exp(u) H e h X 1,..., X k x exp(u) exp 1 (x) = k x i (x)x i H G h H L h (exp(u)) H {(L h (exp(u)); x 1 L 1 h,..., x k L 1 h )} h H H H G H Lie τ : G G G; (g, h) gh 1 τ H τ H τ H : H H H (h 1, h 2 ) H H τ H (h 1, h 2 ) = h 1 h 1 2 G (W ; x 1,..., x n ) V = {z W x i (z) = 0 (k + 1 i n)} H h 1 h 1 2 x 1,..., x k H τ H H (h 1, h 2 ) H H U τ H (U) V U W ; (x, y) xy 1 C (x, y) x i (xy 1 ) U C τ H : U V C τ H C H Lie H G Lie Lie Lie Lie Lie 9.8 Lie G Lie H Lie g, h h = {X g exp tx H(t R), t exp tx H } H Lie h = {X g exp tx H(t R)}

29 Lie h = {X g exp tx H(t R), t exp tx H } h h X h, s R exp sx G (W ; x 1,..., x n ) V = {z W x i (z) = 0 (k + 1 i n)} H exp sx x 1,..., x k H t exp tx H s I exp tx V (t I) I W ; t exp tx C t x i (exp tx) I C t exp tx H C X h h h h = h H Lie H X g exp tx H(t R) t exp tx H h = {X g exp tx H(t R)} 9.9 Lie G Lie H, K Lie h, k H K G Lie Lie h k 9.10 Lie Lie 9.11 G GL(V ) Lie G Lie g 7.2 GL(V ) X g exp G (X) = e X g G, X g Ad G (g)x = d dt (getx g 1 ) = d t=0 dt t=0 etgxg 1 = gxg Lie 10.1 Lie g ρ : g gl(v ) v V h = {X g ρ(x)v = 0} h g Lie a, b R, X, Y h ρ(ax + by )v = aρ(x)v + bρ(y )v = 0, ρ([x, Y ])v = ρ(x)ρ(y )v ρ(y )ρ(x)v = 0 ax + by, [X, Y ] h h g Lie

30 Lie G ρ : G GL(V ) v V H = {g G ρ(g)v = v} H G Lie h H Lie h = {X g dρ(x)v = 0} g, h H ρ(gh 1 )v = ρ(g)ρ(h 1 )v = v gh 1 H H G ρ : G GL(V ) C ρ v : G V ; g ρ(g)v ρ v C H = ρ 1 v (v) G H G Lie ( ) 9.8 X h t R exp tx H ρ(exp tx)v = v t = = d dt t=0 ρ(exp tx)v = d dt etdρ(x) v = dρ(x)v. t=0 dρ(x)v = 0 X g t R ρ(exp tx)v = exp(tdρ(x))v = n=0 exp tx H 9.8 X h 1 n! (tdρ(x))n v = v 10.3 V det : GL(V ) GL(R) = R {0} GL(V ) det tr : gl(v ) gl(r) = R det : GL(V ) GL(R) = R {0} GL(V ) X gl(v ) X : V V λ i (1 i k) p i 6.4 det(e tx ) = k (e λit ) p i = k e p iλ i t. d dt det(etx ) = d t=0 dt k e p iλ i t = t=0 k p i λ i = tr(x) det tr

31 Lie 10.4 V SL(V ) = {g GL(V ) det g = 1} SL(V ) Lie SL(V ) SL(V ) Lie sl(v ) sl(v ) = {X gl(v ) trx = 0} R n Lie SL(n, R), sl(n, R) 10.5 V A : V V R G = {g GL(V ) A(gu, gv) = A(u, v) (u, v V )} G Lie g G Lie g = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )} V V R M 2 (V, R) b, c R, B, C M 2 (V, R) (bb + cc)(u, v) = bb(u, v) + cc(u, v) (u, v V ) bb + cc M 2 (V, R) M 2 (V, R) g GL(V ), B M 2 (V, R) (ρ(g)b)(u, v) = B(g 1 u, g 1 v) (u, v V ) ρ(g)b M 2 (V, R) ρ(g) GL(M 2 (V, R)) g, h GL(V ), u, v V (ρ(gh)b)(u, v) = B((gh) 1 u, (gh) 1 v) = B(h 1 g 1 u, h 1 g 1 v) = (ρ(h)b)(g 1 u, g 1 v) = (ρ(g)(ρ(h)b))(u, v) ρ(gh) = ρ(g)ρ(h) ρ : GL(V ) GL(M 2 (V, R)) u, v V, B M 2 (V, R) g (ρ(g)b)(u, v) C ρ C ρ Lie ρ G = {g GL(V ) ρ(g)a = A} 10.2 G Lie G Lie ρ X gl(v ), B M 2 (V, R), u, v V (dρ(x)b)(u, v) = d dt (ρ(etx )B)(u, v) = d t=0 dt B(e tx u, e tx v) 10.2 = B(Xu, v) B(u, Xv) g = {X gl(v ) dρ(x)a = 0} = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )}. t=0

32 V A V V A O(V ) = O(V ; A) 10.5 O(V ) Lie O(V ) O(V ) Lie o(v ) = o(v ; A) 10.5 o(v ) = {X gl(v ) A(Xu, v) + A(u, Xv) = 0 (u, v V )} R n Lie O(n), o(n) 10.7 O(n) n o(n) n 10.8 V A V SO(V ) = SO(V ; A) = SL(V ) O(V ; A) 9.9 SO(V ) Lie SO(V ) SO(V ) Lie so(v ) = so(v ; A) 9.9 so(v ) = sl(v ) o(v ) = o(v ) R n Lie SO(n), so(n) 10.9 V I : V V ; v 1v V GL R (V ) Lie gl R (V ) 8.12 g GL R (V ) Ad(g)I = gig 1 V GL C (V ) {g GL R (V ) Ad(g)I = I} 9.9 Lie GL C (V ) GL C (V ) Lie gl C (V ) 8.10 dad(x)t = ad(x)(t ) = XT T X (X, T gl R (V )) GL C (V ) = {g GL R (V ) gi = Ig}, gl C (V ) = {X gl R (V ) XI = IX} gl C (V ) V GL C (C n ), gl C (C n ) GL(n, C), gl(n, C) Lie G V G GL C (V ) Lie G Lie g V g gl C (V ) Lie g Lie G G Lie g

33 Lie ρ : G GL C (V ) G I V 8.6 X g I exp(tdρ(x))i 1 = Iρ(exp tx)i 1 = ρ(exp tx) = exp(tdρ(x)) t = 0 Idρ(X)I 1 = dρ(x) dρ(x) gl C (V ) dρ : g gl C (V ) g V det C : GL C (V ) GL C (C) = C {0} GL C (V ) det C tr C : gl C (V ) gl C (C) = C V det C SL C (V ) = {g GL C (V ) det C g = 1} SL C (V ) Lie SL C (V ) SL C (V ) Lie sl C (V ) sl C (V ) = {X gl C (V ) tr C X = 0} C n Lie SL(n, C), sl(n, C) V A V Hermite V A U(V ) = U(V ; A) 10.5 U(V ) Lie U(V ) U(V ) Lie u(v ) = u(v ; A) 10.5 u(v ) = {X gl C (V ) A(Xu, v) + A(u, Xv) = 0 (u, v V )} C n Hermite Lie U(n), u(n) U(n) n u(n) n Hermite V A V Hermite SU(V ) = SU(V ; A) = SL C (V ) U(V ; A) 9.9 SU(V ) Lie SU(V ) SU(V ) Lie su(v ) = su(v ; A) 9.9 su(v ) = sl C (V ) u(v ) C n Hermite Lie SU(n), su(n)

34 31 11 Lie Lie 11.1 Lie G Lie g g Lie h G Lie Lie h dimh = k g G H g = d(l g ) e (α(h)) H g T g (G) k g G H g H G k g X 1,..., X n X 1,..., X k h X i G 1 i k (X i ) g H g (g G) H G k X, Y H f i, g i C (G) X = k f i X i, Y = k g i X i f C (G) [X, Y ](f) = = = k k [f i X i, g j X j ](f) = {f i X i (g j X j (f)) g j X j (f i X i (f))} i,j=1 i,j=1 k {f i (X i g j )(X j f) + f i g j X i X j f g j (X j f i )(X i f) g j f i X j X i f} i,j=1 k {f i (X i g j )X j g j (X j f i )X i + f i g j [X i, X j ]}(f) i,j=1 [X, Y ] = k {f i (X i g j )X j g j (X j f i )X i + f i g j [X i, X j ]}. i,j=1 h g Lie [X i, X j ] h [X, Y ] H H G k Frobenius G e H H g G L g (H) G x H T gx (L g (H)) = d(l g ) x T x (H) = d(l g ) x H x = d(l g ) x d(l x ) e (α(h)) = d(l gx ) e (α(h)) = H gx L g (H) H g H L g 1(H) L g 1(H) H L g 1(H) L g 1(g) = e H L g 1(H) H g 1 L g 1(H) H g 1 H L g 1(H) H L g H L g (H) H H = L g (H) H G H G Lie H G G G 0 Lie 3.1

35 Lie Lie H G 0 τ H : H H G; (g, h) gh 1 C H G τ H (H H) H H H τ H : H H H C H Lie G Lie H H Lie h H G Lie Lie h 9.4 H,H G 7.5 H H U H = {U n n N} = H ( 3.1) H H ι : H H U h H L h 1 ι = ι L h 1 ι L h (U) ι : H H Lie 11.2 Lie G Lie H Lie g, h g G H g = d(l g ) e (α(h)) H G H H 11.3 g Lie g h X g, Y h [X, Y ] h h g 11.4 f : g h Lie kerf g X g, Y kerf f([x, Y ]) = [f(x), f(y )] = 0 [X, Y ] kerf kerf g 11.5 f : G H Lie kerf G Lie ( Lie ) G Lie g kerf g Lie kerdf g G f(g) H Lie Lie df(g) H Lie f kerf G kerf Lie kerf g Lie h 9.8 X g h exp tx kerf (t R) 8.6 f(exp tx) = exp(tdf(x)) f(exp tx) = e (t R) df(x) = 0 X kerdf h = kerdf 8.2 df Lie 11.4 kerdf g G 7.5 exp(g) G 3.1 G = {exp(g) n n N} f(g) = f( {(exp g) n n N}) = {(f(exp g)) n n N} = {(exp(df(g))) n n N} ( 8.2 ) f(g) df(g) Lie

36 V det R : GL R (V ) GL R (R) Lie ( 10.3) SL R (V ) = ker(det R ) GL R (V ) Lie SL R (V ) Lie sl R (V ) gl R (V ) W det C : GL C (W ) GL C (C) Lie SL C (W ) = ker(det C ) GL C (W ) Lie SL C (W ) Lie sl C (W ) gl C (W ) 11.7 Lie G Lie H G H Lie g, h h = {X g exp tx H(t R)} 9.8 h = {X g exp tx H(t R), t exp tx H } exp tx H(t R) X g t exp tx H f : R G; t exp tx C f(r) H H 3.1 H 11.2 H G f H f : R H C H 11.8 G Lie H Lie H H Lie G, H Lie g, h H G h g X g, Y h H G 8.10 exp(ad(exp sx)(ty )) = exp(sx) exp(ty ) exp(sx) 1 H (s, t R) Ad(exp sx)y h d Ad(exp sx)y ds = [X, Y ] ( 8.10) s=0 [X, Y ] h h g 11.9 G Lie H Lie G, H Lie g, h H G h g 11.8 H G h g h g 7.5 G U g

37 Lie Lie 0 V exp : V U exp(v h) H X g, Y h (exp X)(exp Y )(exp X) 1 = exp(ad(exp X)Y ) 8.6 Ad 8.10 Ad(exp X)Y = e adx 1 Y = n! (adx)n Y h (exp X) exp(v h)(exp X) 1 H 3.1 (exp X)H(exp X) 1 = = n=0 (exp X) exp(v h) n (exp X) 1 n=1 ((exp X) exp(v h)(exp X) 1 ) n H n=1 G = {U n n N} g G ghg 1 H H G g Lie ker(ad) = {X g Y g [X, Y ] = 0} g ( 11.4 ) f, g Lie Lie Lie df = dg f = g f, g Lie G Lie H Lie X g 8.6 f(exp X) = exp(df(x)) = exp(dg(x)) = g(exp X) f g exp(g) G = {exp(g) n n N} f g G Lie G Lie g G Ad G Lie Lie g G Z ( 8.10, 8.11) Z ker Ad g ker Ad d(l g Rg 1 ) g L g Rg 1 G g Z Z = ker Ad 11.5 Z G Lie Z g Lie ker dad = ker ad ( 8.10) g

38 Lie Lie G Lie Lie g G g 7.8 g g g G G G 12 Lie 12.1 U(n) exp : u(n) U(n) U(n) U(n) u g U(n) a 1,..., a n C a 1 g 1 ug =... g 1 ug U(n) a 1 = = a n = 1 θ 1,..., θ n R a i = e 1θ i (1 i n) a 1 1θ1 u = g... g 1 = exp Ad(g).... 1θn a n a n θ1... 1θn u(n) g U(n) 1θ1 Ad(g)... 1θn u(n) U(n) u(n) U(n) 12.2 SU(n) exp : su(n) SU(n) SU(n)

39 Lie 12.1 SU(n) u g U(n) θ 1,..., θ n R ( ) 1θ1 u = g exp... 1θn g 1. det u = 1 θ θ n = 2πk (k Z) e 1θ 1 = e 1(θ 1 2πk) θ 1 θ 1 2πk ( ) θ θ n = θ1... su(n) 1θn g U(n) Ad(g) 1θ1... su(n). 1θn SU(n) su(n) SU(n) 12.3 SO(n) exp : so(n) SO(n) SO(n) SO(n) U(n) 12.1 SO(n) u g U(n) θ 1,..., θ n R e 1θ 1 u = g... g 1. e 1θ n u u u g e 1θ 1 e 1θ 1... e 1θ k ( ) u = g e 1θ k g 1 ε 2k+1... εn

40 37 θ i / πz (1 i k), ε j = ±1 (2k + 1 j n) g = [g 1... g n ] ug 2i 1 = e 1θ i g 2i 1 (1 i k) u uḡ 2i 1 = e 1θiḡ 2i 1 (1 i k) g 2i ḡ 2i 1 g U(n) ( ) ε j R g j R n det u = 1 ε j = 1 ε j g ε 2k+1 = = ε 2l = 1, ε 2l+1 = = ε n = 1 1 i k h 2i 1 = 1 2 (g 2i 1 + ḡ 2i 1 ), h 2i = 1 2 (g 2i 1 ḡ 2i 1 ) h j = g j (2k + 1 j n) h = [h 1... h n ] U(n) h h O(n) 1 i k uh 2i 1 = 1 2 (e 1θ i g 2i 1 + e 1θiḡ 2i 1 ) = cos θ i h 2i 1 sin θ i h 2i uh 2i = 1 2 (e 1θ i g 2i 1 e 1θiḡ 2i 1 ) = sin θ i h 2i 1 + sin θ i h 2i θ i = π [ (k + 1 i l) ] R i = cos θ i sin θ i sin θ i cos θ i (1 i l) u = h R 1... R l h 1. X i = [ 0 θ i θ i 0 ] (1 i l) exp X i = R i X 1... u = exp Ad(h) X l

41 Lie X 1... Ad(h) X l o(n) = so(n) SO(n) so(n) SO(n) 12.4 O(n) 2 SO(n) {g O(n) det g = 1} det(o(n)) = {±1} SO(n) {g O(n) det g = 1} O(n) 12.3 SO(n) det h = 1 h O(n) {g O(n) det g = 1} = L h (SO(n)) SO(n) L h (SO(n)) O(n) O(n) = SO(n) L h (SO(n)) O(n) 12.5 GL(n, C) SL(n, C) X gl(n, C) (i, j) X ij T (n, C) = {X gl(n, C) X ii > 0(1 i n), X ij = 0(i > j)} T (n, C) gl(n, C) T (n, C) X T (n, C) det X = n X ii > 0 X GL(n, C) T (n, C) GL(n, C) T (n, C) GL(n, C) P : U(n) T (n, C) GL(n, C); (a, X) ax P P g GL(n, C) g = [g 1... g n ] g i C n g 1,..., g n C n C n Hermite g 1,... g n Gram-Schmidt b 1 = g 1, a 1 = 1 b 1 b 1 b k, a k (k 2) k 1 b k = g k g k, a i a i, a k = 1 b k b k. a 1,..., a n C n a k g 1,..., g k g k 1/ b k > 0 ( ) [a 1... a n ] = [g 1... g n ]X

42 39 X T (n, C) a = [a 1... a n ] a U(n) a = gx g = ax 1 X 1 T (n, C) g = P (a, X 1 ) P 12.1 U(n) T (n, C) GL(n, C) detx. S : GL(n, C) SL(n, C); X X S X SL(n, C) S(X) = X S(GL(n, C)) = SL(n, C) SL(n, C) 12.6 GL(n, R) 2 GL + (n, R) = {g GL(n, R) det g > 0}, {g GL(n, R) det g < 0} SL(n, R) 12.5 det : GL(n, R) GL(1, R) = R {0} GL + (n, R) GL(n, R) GL + (n, R) GL(n, R) Lie T (n, R) = gl(n, R) T (n, C) T (n, R) gl(n, R) T (n, R) T (n, R) GL + (n, R) P (SO(n) T (n, R)) = GL + (n, R) 12.5 g GL + (n, R) a O(n) X T (n, R) a = gx det g > 0, det X > 0 det a = 1 a SO(n) g = ax 1 X 1 T (n, R) P (SO(n) T (n, R)) = GL + (n, R) 12.3 SO(n) T (n, R) GL + (n, R) det 1 ({t R t > 0}) = GL + (n, R), det 1 ({t R t < 0}) = {g GL(n, R) det g < 0} GL(n, R) 2 GL(n, R) ST (n, R) = T (n, R) SL(n, R) S(T (n, R)) = ST (n, R) T (n, R) ST (n, R) g SL(n, R) a = gx, a SO(n), X T (n, R) det g = 1 det X = 1 P (SO(n) ST (n, R)) = SL(n, R) SO(n) SL(n, R)

43 Lie 12.7 V n B R (V ) = {(v 1,..., v n ) V n v 1,..., v n V } V B R (V ) B R (V ) V n GL R (V ) B R (V ) 2 n W W B C (W ) B C (W ) W n GL C (W ) B C (W ) V e 1,..., e n 1 T : gl R (V ) V n ; g (g(e 1 ),..., g(e n )) T T (GL R (V )) = B R (V ) B R (V ) V n GL R (V ) 12.6 B R (V ) 2 W T B C (W ) W n GL C (W ) 12.5 B C (W ) 12.8 V B R (V ) 1 V 12.7 V n V 2 u 1,..., u n v 1,..., v n V X GL(n, R)( (u 1... u n ) = (v 1... v n )X) GL + (n, R) n V 2 u 1,..., u n v 1,..., v n V X O(n)( (u 1... u n ) = (v 1... v n )X) SO(n)