Dynkin Serre Weyl

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1 Dynkin Naoya Enomoto paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie Lie ( ) Killing form sl 2 (C) ( ) Lie Cartan ( ) Killing ( ) sl Dynkin Dynkin henon@s00x0427.mbox.media.kyoto-u.ac.jp 1 1

2 Dynkin Serre Weyl A n 1 sl n B n so 2n C n sp 2n D n so 2n G E 6, E 7, E 8, F II Dynkin Dynkin 32 III Dynkin 32 7 SO(3) 32 IV Dynkin 32 V Quiver Dynkin 32 VI Painlevé Dynkin 32 2

3 0 Introduction 3

4 I ( ) Lie Dynkin Dynkin Lie Dynkin 1 ( ) Lie ( ) Lie ( ) Killing sl 2 (C) 1.1 Lie ( ) Lie Lie 1.1. ( ) g Lie g [, ] : g g C (L1)[ ] [x, y] = [y, x] (L2)[Jacobi ] [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (x, y, z g) (1.1) Lie (1) g 1, g 2 Lie f([x, y]) = [f(x), f(y)] f : g 1 g 2 Lie V End(V ) [A, B] := AB BA Lie f : g End(V ) Lie Lie g (2) g ad : g End(g); g ad g = [g, ] (1.2) g g (3) Lie g h [h, h] h Lie g a [g, a] a a g (4) g [x, y] = 0 Lie Lie Lie Lie Lie 1.3. Lie Lie (1) Lie (2) (3) 3 (4) Lie g g = a 0 a 1 a 2 a r = {0} a i /a i+1 Lie g Lie 4 Lie g τ(g) g 4

5 ( 5 )Lie 1.4. gl n (k) Lie sl n (k) := {x gl n (k) tr(x) = 0} (1.3) so n (k) := {x gl n (k) t x + x = 0} (1.4) sp 2m (k) := {x gl 2m (k) t xj + Jx = 0} (n = 2m) (1.5) J = ( 0 E m E m 0 ) 1.2 Killing form g Killing g ( ) 1.5. Lie g ( ), g- [x, y], z = x, [y, z] Rad, := {x g g y x, y = 0} Rad, = {0}, Remark 1.6., Rad, (ad, g) g Lie Rad, = g {0}, (g, V ) Lie B V (x, y) := tr(π(x)π(y)) (x, y g) g ad B(x, y) := tr(ad x ad y ) (x, y g) (1.6) g Killing 1.8. Cartan s criterion Lie g (1) g (2) Killing B Cartan Lie Remark Schur 1.9. g Lie g Killing 5 n 5

6 [ ] g 2 0 g Remark g = g g g Schur Schur Schur (g, V ) End g (V ) = C Id V g End g (g) = C Id g [ ] T End(V ) T λ v( 0) 1 T v = λv T := T λid V T End(V ) Ker T V V {0} 0 v Ker T Ker T = V T = 0 T = λid V g 1.3 sl 2 (C) sl 2 (C) Lie Lie sl 2 sl 2 = {X M 2 (K) tr X = 0} Lie 3 ( ) ( ) ( e =, f =, h = ) (1.7) [h, e] = 2e, [h, f] = 2f, [e, f] = h (1.8) sl 2 (C) ( ) sl 2 (C) (sl 2, V ) h v hv = cv ev, fv c + 2, c 2 (sl 2, V ) (1) V h ev 0 = 0 v 0 ( ) (2) v 0 ( ) n n Z 0 dim V = n + 1 sl 2 [ ] [ ] h(ev) = ([h, e] + eh)v = 2ev + cev = (c + 2)ev, h(fv) = ([h, f] + fh)v = 2fv + cv = (c 2)v h v {v, ev, e 2 v, } V 0 e k+1 v = 0 v 0 := e k v 6

7 ev 0 = 0 h n {v 0, fv 0, f 2 v 0, } r f r+1 v 0 = 0 f r v 0 0 {f i v 0 i r} sl 2 - V V = r i=0 C(f i v 0 ) v i = f i v 0 0 = (ef r+1 r r+1 e)v 0 ( f r+1 v 9 = fv r = 0, ev 0 = 0) = [e, f r+1 ]v 0 = (r + 1)(h r)v 0 ( [e, f k ] = k(h (k 1)) ) = (r + 1)(n r)v 0 v 0 0 (r + 1)(n r) = 0 r 0 r n = r dim V = r + 1 = n + 1 (1)(2) [ ] Remark [e, f k ] = k(h (k 1)) hf = f(h 2), (h + 2)f = fh [e, f n ] = [e, f]f n 1 + f[e, f]f n f n 1 [e, f] = hf n 1 + fhf n f n 1 h ( [e, f] = h) = hf n 1 + (h + 2)f n (h + 2(n 1))f n 1 ( f ) = (nh + n(n 1))f n 1 = nf n 1 (h 2(n 1) + (n 1)) ( f ) = nf n 1 (h (n 1)) Remark sl 2 h e Cv 0 f Cv 1 fig. Cv 2 sl 2 Cv r Remark

8 m( ) m( ) e f k m k m V m k 0 k 1 k 2 k 3 V 3 V 2 V 1 V 0 V [0] V [1] V [2] V [3] V [m] V [m] V = V m, V = m Z V k m = dim V [m] m=0 V [m] dim V 0 = k m, dim V 1 = m:even m:odd m k m 2 ( ) Lie Lie Killing sl Cartan ( ) Lie Lie 2.1. g Lie h Cartan 2 (1) h (2) g ad H (H h) Remark 2.2. Cartan 2 Cartan h 1, h 2 6 ϕ Inn(g) ϕ(h 1 ) = h 2 Cartan g 6 8

9 ad(h) gl(g) g ad(h) (root space decomposition) x g ad(h) [h, x] = λ h x λ h = α(h) α : h h λ h C α h (h ) α g α = {x g [h, x] = α(h)x h h} 2.3. Φ = {α h α 0, g α {0}} α Φ g g α α g ad(h) ( ) g = g 0 g α Remark 2.4. g 0 X g [h, X] = 0 ( h h) h g 0 h h = g 0 ( ) g = h g α α Φ α Φ 2.2 ( ) Killing Killing g Lie Cartan h g ( ) g = h g α α Φ 2.5. (1) [g α, g β ] g α+β (2) dim g α = dim g α (3) Killing B g α g α := {x g B(x, y) = 0 y g α } g α = β α g β (4) B h B h h 9

10 [ ] (1) B h h, x g α, y g β [h, [x, y]] = [[h, x], y] + [x, [h, y]] ( Jacobi ) = [α(x)x, y] + [x, β(h)y] ( x g α, y g β ) = (α(h) + β(h))[x, y] = (α + β)(h)[x, y] (2),(3) (1) x g α, y g β ad(x) ad(y)g γ = [x, [y, g γ ]] g α+β+γ g ad(x) ad(y) α + β 0 0 β α B(x, y) = tr(ad(x) ad(y)) = 0 g β g α g α β α g β dim g = n B dim g α = n dim g α g α β α g β dim g α β α dim g β = n dim g α dim g α dim g α α α (2) dim g α = dim g β (3) (4) (3) α = 0 β α h = α Φ g α h h = {0} (4) Remark 2.6. (1) α, β Φ α + β Φ g α+β = 0 Hom(V V, C) = Hom(V, V ) {V } {V = V } 10

11 B h h φ h 1 t φ h B(t φ, h) = φ(h) Killing sl (1) Φ h dim h = j Φ j 1 (2) α Φ α Φ (3) α Φ, x g α, y g α [x, y] = B(x, y)t α (4) α Φ dim([g α, g α ]) = 1 t α (5) α Φ α(t α ) = B(t α, t α ) 0 (6) α Φ 0 x α g α y α g α x α, y α, h α = [x α, y α ] 3 Lie sl 2 (C) Lie 2t α (7) h α = B(t α, t α ), t α = t α α(h α ) = 2 [ ] (1) Φ h g C g C Φ β 1,, β k h 1,, h k h h 1,, h j x 1 β i (h 1 ) + + x j β i (h j ) = 0 (1 i k) k(< j) x 1,, x j C = x 1 h x j h j 0 C h α(c) = 0 ( α Φ) g g g g = h + α Φ g α [C, g] = [C, h] + α [C, g α ] = [C, h] + α α(c)g α = 0 [C, g] = 0 H g (2) dim g α = dim g α α Φ dim g α 0 dim g α 0 α Φ (3) α Φ, x g α, y g α h h B(h, [x, y]) = B([h, x], y) ( B ) = α(h)b(x, y) ( x g α ) = B(t α, h)b(x, y) ( t α ) = B(B(x, y)t α, h) [x, y] g 0 = h, B(x, y)t α h [x, y] B(x, y)t α h h B h h [x, y] = B(x, y)t α (4) (3) [g α, g α ] t α [g α, g α ] 0 0 x g α B(x, g α ) = 0 B(x, g) = 0 B B(x, y) 0 y g α [x, y] = B(x, y)t α 0 (5) α(t α ) = 0 x g α, y g α [t α, x] = α(t α )x = 0, [t α, y] = α(t α )y = 0 x, y (4) B(x, y) 0 B(x, y) = 1 [x, y] = t α S = x, y, t α g 3 Lie 11

12 [S, S] = t α [g, DS] = 0 S Lie ad(t α ) t α h Cartan ad(t α ) ad(t α ) = 0 t α = 0 α = 0 (6) 0 x α g α y g α B(x α, y α ) = 2 B(t α, t α ) (5) B(t α, t α ) 0 (4) B(x, y) 0 (3) h α = 2t α B(t α, t α ) [x α, y α ] = B(x α, y α )t α = h α [h α, x α ] = 2 α(t α ) [t α, x α ] = 2 α(t α ) α(t α)x α = 2x α [h α, y α ] = 2y α S α = x α, y α, h α = sl 2 (7) h α α(h α ) = 2 α(t α ) α(t α) = 2 h h B(t α, h) = α(h) B t α = t α B(t α + t α, h) = α(h) α(h) = ( ) sl 2 g sl S α = x α, y α, h α = sl 2 (1) α Φ dim g α = 1 0 x α x α x α [x α, y α ] = h α y α g α (2) α φ cα Φ c = ±1 (3) α, β Φ β(h α ) Z β β(h α ) Φ (4) α, β, α + β Φ [g α, g β ] = g α+β (5) α, β Φ β ±α p, q Z β pα, β + qα Φ p i q i β + iα Φ β(h α ) = p q (6) g Lie {g α } α Φ [ ] (1),(2) (3) (6) sl 2 (1),(2) STEP.1 sl 2 M 12

13 M = h c 0 g cα x α, y α, h α S α = sl2 X, Y, H sl 2 ad xα, ad yα, ad hα g sl 2 - M g sl 2 - m M m = j m j (m j g cj α c j = 0 h ) ad(h α )(m) = [h α, m] = j [h α, m j ] = j c j α(h α )m j M ad(x α )(m) = [x α, m] = j [x α, m j ] g α+cj α ad(y α )(m) = [y α, m] = j [y α, m j ] g α+cj α M sl 2 STEP.2 2c Z ad hα v g cα ad(h α )(v) = [h α, v] = cα(h α )v = 2cv 2c Z STEP.3 (1),(2) V m, V [m], k m M Remark1.14 h V 0 = h g k/2 = V k V ±2 = g ±α h α h V [0] h h α h t α h α α(h) = B(t α, h) = 0 α(h) = 0 [x α, h] = α(h)x α = 0 [y α, h] = 0 h 1 h V [0] dim h = l h α h - l 1 h α h l 1 V [0] = k 0 h α, x α, y α = S α V [2] 1 V [2] = k 2 l = dim h = dim V 0 = m:even k m k 0 = l 1, k 2 = 1, k m = 0 (m : even, m 4) dim g α = dim V 2 = k 2 + k 4 + = 1 dim g 2α = dim V 4 = k 4 + k 6 + = 0 dim g 3α = dim V 6 = k 6 + k 8 + = 0 2α, 3α, 2α, 3α, 1 2α 2 α 1 2 α 0 = dim g (1/2)α = dim V 1 = m:odd k m 13

14 (1),(2) k m = 0 (m : odd) (3) (6) sl 2 β ±α K = i Z g β+iα sl 2 - g β+iα {0} i dim g β+iα = 1 (1) g β+iα 0 β(h α ) + 2i i Z 0 1 i 1 i 2 ad hα m, m 2,, m p, q β + iα β pα,, β + qα (β(h α ) + 2q) = β(h α ) 2p β(h α ) = p q (5) p q p q β β(h α )α = β (p q)α (3) (4) dim g α+β = 1 ad xα (g β ) {0} (6) h [x α, y α ] = h α 2.4 g Lie h Cartan g ( ) g = h g α Φ h α 1,, α r Φ h g Killing form B (α, β) := B(t α, t β ) h (, ) t α α(h) = B(t α, h) ( h h) h β Φ k β = c i α i α Φ 2.9. c i Q Proof k (β, α j ) = c i (α i, α j ) (j = 1, 2,, k) 14

15 β(h α ) Z 2(β, α j ) k (α j, α j ) = 2(α i, α j ) (α j, α j ) c i (j = 1, 2,, k) ( ) 2(β, α) (α, α) = 2β(t α) B(t α, t α ) = β(h α) Z ( ) B α 1,, α k h ( ) c i Q ad h h 0- g α α(h) B(h, h ) = tr(ad h ad h ) = α Φ α(h)α(h ) γ, δ h (γ, δ) = α φ α(t γ )α(t δ ) = α Φ(α, γ)(α, δ) β Φ (β, β) = α Φ(α, β) 2 0 E = α 1,, α k R (γ, γ) 0 ( γ E) (γ, γ) = 0 α Φ (α, γ) = 0 B γ = 0 k E (, ) α 1,, α k Φ E 2.5 g ( ) Lie g Cartan h ad(h) g g = h α Φ g α g α = {x g [h, x] = α(h)x h h} g 0 = h Φ = {α h α 0, g α {0}} h g g Φ g Killing h h h {α 1,, α l } h R Φ h R C αβ = 2(β, α) (α, α) 15

16 C αβ = β(h α ) C αβ Z β C αβ α Φ C αβ α σ α Φ h R Φ 4 (R1) Φ 0 / Φ (R2) α Φ, c R cα Φ c = ±1 (R3) α, β Φ α Φ σ α Φ (R4) E R Φ C αβ = α, β := 2(α, β) (α, α) Z E Φ 4 ( ) Lie g Cartan h Φ Dynkin Lie E R l (, ) l E E ( 7 ) 3.1. E Φ E ( ) (R1) Φ 0 / Φ (R2) α Φ, c R cα Φ c = ±1 (R3) α, β Φ α Φ σ α Φ (R4) E R Φ C αβ = α, β := 2(α, β) (α, α) Z Remark 3.2. E (, ) 0 α E α E P α β E P α σ α (β) σ α (β) = β 2(α, β) (α, α) α = β C αβα E = Rα P α β = cα + γ σ α (β) = cα + γ = β 2cα (α, β) = c(α, α) c = (α, β)/(α, α) C αβ 7 Lie E 16

17 4 1 1 E = R 2 E = R 2 A 1 A 1 A 1 A E Φ Φ Φ 1, Φ 2 Φ 1 Φ 2 = φ, Φ 1 Φ 2 = Φ, Φ 1 Φ A 2 B 2 (= C 2 ) 2 G Dynkin 8 G 2 17

18 3.2 α, β Φ (α, β) 0 α, β α, β α β ( ) α α (α, β) = α β cos θ (cos θ) 2 = (α, β) > 0 ( ) (α, β)2 α 2 β 2 = C αβc βα 4 C αβ, C βα 0 cos θ 1 C αβ C βα = 1, 2, 3 ( ), ( ) C αβ C βα α, β θ C αβ C βα (cos θ) 2 θ C αβ C βα β / α 1 1/4 π/ /2 π/ /4 π/ β β β π/3 π/4 α α π/6 α 3.4. α, β Φ (1) (α, β) > 0 α β Φ (2) (α, β) < 0 α + β Φ (3) Z p, q 0 β + iα Φ q i p [ ] (1) (α, β) > 0 α β C βα = 1 (R3) Φ σ β (α) = α C βα β = α β 18

19 α > β β α Φ (R2) α β = (β α) Φ (2) (1) β β (3) I = {i Z β + iα Φ} Φ E I p, q β Φ 0 I p, q 0 p, q p, q I = {i Z q i p, β + iα / Φ} r, s q < s r < p, β + rα / Φ, β + (r + 1)α Φ, β + sα / Φ, β + (s 1)α Φ β + rα = (β + (r + 1)α) α, β + sα = (β + (s 1)α) + α (1)(2) (α, β + (r + 1)α) 0, (α, β + (s 1)α) 0 (r s + 2)(α, α) 0 r s + 2 > 0, (α, α) > 0 {β + iα q i p} β α (1) C αβ = q p (2) β α- 4 [ ] (1) α σ α Φ β α- σ α (β + pα) = β qα σ α (α) = α C αβ p = q C αβ = q p (2) β = β + pα C αβ = C αβ + 2p = p + q C αβ 3 α- p + q + 1 p + q + 1 = C αβ α β α- β 2α β α β β α β β α α β 3α β 2α β α β α 19

20 3.3 (E, Φ) 3.6. v V α Φ (v, α) 0 v Remark 3.7. ( ) E\ P α α Φ Φ (E, Φ) v 3.8. (1) Φ + = Φ + (v) := {α Φ (v, α) > 0} Φ Φ = Φ + Φ = Φ + Φ Φ +, Φ (2) α Φ + α = β 1 + β 2 (β i Φ + ) α Φ + Π = Π(Π) (3) Φ Π Φ + Π = Π(Φ + ) Π Φ 3.9. S Φ Φ 2 (i) S E (ii) β Φ β = m α α α S m α Z m α 0 m α (1) Π Φ α β Π (α, β) 0 (2) S Φ + α β S (α, β) 0 S [ ] (1) (α, β) > 0 α β, (α β) = β α Φ (α, v) = (β, v) (α β, v) = 0 α β Φ v (α, v) (β, v) v (α, v) > 0, (β, v) > 0 (α β, v) = (α, v) (β, v), (β α, v) = (β, v) (α, v) (α, v), (β, v) α β Φ + β α Φ + α = (α β) + β, β = α + (β α) α, β Π (2) α S C αα = 0 (C α Q) λ = C C α 0 αα, µ = C C α 0 αα λ + µ = α S C αα = 0 0 (λ, λ) = (λ, µ) = C α 0,C α 0 C α ( C α )(α, α ) 0 20

21 C α C α α α (α, α ) 0 (λ, λ) = 0 λ = µ = 0 Φ + = Φ + (v) C α (v, α) = (v, λ) = 0 = (v, µ) = C α (v, α) C α 0 C α 0 α S Φ + (v, α) > 0 α S C α = 0 [ 3.9 ] Π v Π(Φ + (v)) Π β Φ + β Π β = β 1 + β 2 (β i Φ + (v)) (v, β) = (v, β 1 ) + (v, β 2 ) > (v, β i ) (i = 1, 2) (v, β i ) > 0 β = m α α (0 m α Z) α Π β Φ β Φ + (R1) Φ E Φ Π Π E (i) (ii) S Φ (i),(ii) E S dual S v = u S u α S (v, α) > 0 (ii) β Φ (v, β) = m α (v, α) 0 α S v (ii) Φ + (v) = {β Φ β = α S M αα m α 0} S Φ + (v) S α = β 1 +β 2 (β i Φ + ) β 1, β 2 S α = β 1 + β 2 α 0 α β i α 0 S Π(Φ + (v)) dim E S = Π(Φ + (v)) 4 Dynkin 4.1 Dynkin Φ Π = {α 1,, α l } (R3) C ij = 2(α i, α j ) (α i, α i ) Z (i, j)- l C Cartan Cartan (C1) C ii = 2 (C2) i j C ij = 0, 1, 2, 3 (C3) C ij = 0 C ji = 0 ( C ij = 2 C ji = 1 C ij = 3 C ji = 1 ) 21

22 Cartan Dynkin (D1) l α 1,, α l (D2) i j α i α j max( C ij, C ji ) (D3) i j C ij 2 ( C ij 2) α i α j α j α i Dynkin Lie 4.2 = {α 1,, α l } E [ ] := {ε 1,, ε l } (ε i = α i / α i ) 4.1. E Γ ε- (A1) {ε 1,, ε l } (A2) ε i, ε j = 0, 1/2, 1/ 2, 3/2 ε i, ε j (i j) π(1 1/m ij ) (m ij = 2, 3, 4, 6) ε- Γ Dynkin ε- Γ = [ ] Dynkin ε ε- [A l ] [B l ] = [C l ] [D l ] [E 6 ] [E 7 ] [E 8 ] [F 4 ] [G 2 ] 4.3. (1) Γ ε- Γ Γ Γ ε- 22

23 (2) Γ = {ε 1,, ε n } ε- ε i, ε j = 0 (i j) (i, j) n (3) ε- Γ = η 1,, η m } η i, η i+1 0 (1 i m 1), η m, η 1 = 0 (1) n (2) ε = ε i 0 < ε = n + 2 i<j ε i, ε j (A1) ε i, ε j = 0 ε i, ε j 1/2 (A2) ε i, ε j = 0 (i, j) (i < j) n n n n < n (3) (2) ε- ε i, ε j = 0 (i, j) (i < j) (2) (1) ε- ε- ε (1) Γ ε- ε Γ ε 3 (2) ε ε- ε 3 (3) ε- Γ Γ = [G 2 ] ε Γ {η Γ ε, η = 0, ε η} = {η 1,, η k } (3) η i, η j = 0 (i j) ε, η i, η j {η 1,, η k } c i = η i, ε, k ε = ε c i η i η i, ε = 0 (1 i k) (A1) ε 0 1 = ε 2 = ε 2 + k c 2 i ε- η i ε 4c 2 i k 4 c 2 i = 4(1 ε 2 ) < 4 (1) ε- 6 (2),(3) 4.5. ε- Γ [A k ] Γ ε 1 ε 2 ε k 23

24 k ε = ε i Γ = Γ Γ Γ {ε} ε- ε- [A k ] 1 ε- ε 2 = k + 2 ε i, ε j = k (k 1) = 1 1 i<j k ε Γ {ε} (A1) η Γ η, ε i 0 i (1 i k) 2 2 η 3 { 0 η, ε i = η, ε i i (A2) [G 2 ] ε- Γ Γ Γ 2 ε- Γ, Γ Γ Γ Γ 9 [A k ] ε- Γ, Γ [A k ] Γ 4.6. ε- Γ [B l ] (l 2), [F 4 ] Γ 9 24

25 ε 1 ε p η q η 1 ε = ε = p iε i, η = p i 2 p q iη i i(i + 1) = 1 p(p + 1) 2 η = 1 q(q + 1) 2 ε, η = pε p, qη q = pq 2 p 2 q 2 = ε, η 2 < ε 2 η 2 = 2 p(p + 1)q(q + 1) 4 ε = η 2pq < (p + 1)(q + 1) 2 (p 1)(q 1) < 2 { p q = 2 p = q = 2 [B l ] [F 4 ] Γ Γ ε- [A l ] Γ 2 [A k ] 4 1 [A k ] 4.7. ε- [D l ] (l 4), [E l ] (l = 6, 7, 8) Γ η 1 ψ η q 1 ε 1 ε p 1 ξ r 1 ξ 1 25

26 p 1 q 1 r 1 ε = iε i, η = iη i, ξ = iξ i ε, η, ξ ε = ε/ ε, η = η/ η, ξ = ξ/ ξ, c 1 = ε, ψ, c 2 = η, ψ, c 3 = ξ, ψ, ψ = ψ c 1 ε c 2 η c 3 ξ ε, η, ξ, ψ ε = η = ξ = 1 p 1 ε 2 = 1 = ψ 2 = ψ 2 + c c c 2 3 p 2 i 2 i(i + 1) = 1 p(p 1) 2 ε, ψ = (p 1)ε p 1, ψ = 1 (p 1) 2 c 2 1 = c 2 2 = 1 2 ε, ψ 2 ε 2 = 1 ( 1 1 ) 2 p ( 1 1 ) c 2 3 = 1 ( 1 1 ) q 2 r {( ) ( ) ( ) } < 1 2 p q r 1 p + 1 q + 1 r > 1 p q r( 2) 3/r r < 3 r = 2 q < 4 q = 2, < 1 p + 1 q 2 q q = 2 p Γ [D l ] (l = p + 2) q = 3 1/p > 1/6 p < 6 p = 3, 4, 5 [E 6 ], [E 7 ], [E 8 ] (p, q, r) = (3, 3, 2), (4, 3, 2), (5, 3, 2) Dynkin [B l ], [F 4 ], [G 2 ] Dynkin [B l ] 2 B l, C l Dynkin (B 2 = C 2 )[F 4 ], [G 2 ] Dynkin 26

27 5 5.1 Dynkin Cartan Lie g Cartan h Cartan Lie Lie g Cartan h Lie 5.1. k Lie g( ) Weyl Lie Dynkin 5.2. (1) g Lie (2) g (3) g (4) Dynkin Dynkin Cartan Dynkin Dynkin Cartan Killing Cartan Dynkin Dynkin Dynkin 5.3. Dynkin Lie 27

28 5.2 Serre Lie Serre 5.4. Π = {α 1,, α l } h i := α h [e i, f i ] = h i e i g αi, f i g αi Lie g Serre g {h i, e i, f i 1 i l} [h i, h j ] = 0 (5.1) [h i, e j ] = a ij e j, [h i, f j ] = a ij f j (5.2) [e i, f j ] = δ ij h i (5.3) (ad ei ) aij+1 (e j ) = 0, (ad fi ) aij+1 (f j ) = 0 (i j) (5.4) 2 Serre 5.3 Weyl 5.5. (E, Φ) W = W (Φ) := σ α α Φ Φ Weyl 5.6. Weyl (1) W (2) W (Π) = Φ E W Weyl (3) W = σ αi α i Π (4) W E 6 Lie 4 A l, B l, C l, D l 5 E 6, E 7, E 8, F 4, G 2 Lie Lie A l B l C l D l sl l+1 (k) so 2l+1 (k) sp 2l (k) so 2l (k) Caylay Jordan l = 1 A 1 sl 2 Dynkin 1 D 1 so 2 (C) = {X M 2 (R) t X + X = 0} C 2 Lie 1 D 1 Dynkin B 1,C 1 so 3, sp 2 3 sl 2 = so3 = sp2 28

29 2 D 2 so 4 so 4 = sl2 sl 2 2 D 2 = A 1 A 1 B 2, C 2 so 5, sp 4 so 5, sp 4 Dynkin = 3 D 3 so 6 so 6 = sl4 A 3 Dynkin = 6.1 A n 1 sl n A Lie Lie 10 A n 1 Lie sl n = {X M n (C) tr X = 0} Cartan h = {X sl n X } E ij (i, j)- h = diag(h 1,, h n ) trh = 0 h h n = 0 [h, E ij ] = (h i h j )E ij (i j) i j α ij h α ij : h h = diag(h 1,, h n ) h i h j C 10 B,C,D 29

30 E ij C g αij h h g 0 sl n sl n = h E ij C i j h = g 0, g αij = E ij C Φ = {α ij i j} A n 1 Cartan fig. Cartan Killing h = diag(h 1,, h n ), h = diag(h 1,, h n) B(h, h ) = tr(ad h ad h ) = i j (h i h j )(h i h j) ( ) n = 2(n 1) h i h i 2 h i h j i j ( n n ) ( n = 2n h i h i 2 h i = 2n h i ) n h i h i ( tr h = tr h = 0 ) α ij (h) = h i h j α ij h t αij Killing h i h j = α ij (h) = B(t αij, h) = 2n n (t αij ) k h k k=1 t αij = 1 diag(0,, 1,, 1,, 0) 2n (α ij, α ij ) = B(t αij, t αij ) = 1 4n 2 (2n 2) = 1 n 30

31 h αij = 2t αij B(t αij, t αij ) = 2n 1 diag(0,, 1,, 1,, 0) = diag(0,, 1,, 1,, 0) 2n ε i = α i,i+1 Π = {ε i 1 i n 1} Π h i < j α ij = ε i + ε i ε j 1 i < j Φ + = {α ij i < j} Cartan (ε i, ε j ) = B(α i,i+1, α j,j+1 ) = 1 2n 4n 2 [diag(e ii E i+1,i+1 ) diag(e jj E j+1,j+1 )] ([ ] ) = 1 2 i = j 2n 1 i = j ± 1 0 otherwise C ij = 2(ε i, ε j ) (ε i, ε i ) = 2 i = j 1 i = j ± 1 0 otherwise Cartan Dynkin A n B n so 2n C n sp 2n 6.4 D n so 2n 6.5 G E 6, E 7, E 8, F 4 31

32 II Dynkin Dynkin III Dynkin 7 SO(3) IV Dynkin V Quiver Dynkin VI Painlevé Dynkin 32

Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ),

Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ), 1 1 1.1,,. 1.1 1.2 O(2) R 2 O(2).p, {0} r > 0. O(3) R 3 O(3).p, {0} r > 0.,, O(n) ( SO(n), O(n) ): Sym 0 (R n ) := {X M(n, R) t X = X, tr(x) = 0}. 1.3 O(n) Sym 0 (R n ) : g.x := gxg 1 (g O(n), X Sym 0