( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h
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- くにひと みつだ
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1 Lie (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h g) f (2.5) g f h h g f f a b g h g g f c d h (2.6) 1
2 ( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) hom-set 2.2 a a a (endomorphism) f : a b f : b a f : b a f f = 1 b f f = 1 a, (2.10) f f (right inverse) f f (left inverse) f = 1 a f = (f f) f = f (f f ) = f 1 b = f (2.11) f b a 1 1 a f b b a f = f f (inverse) f = f 1 f (invertible) (isomorphism) a b (isomorphic) a = b a a a (automorphism) Set f 1 : a 1 a 2, f 2 : a 2 a 4, f 3 : a 1 a 3, f 4 : a 3 a 4 f f 2 f 1 = f 4 f 3 (2.12) 2
3 (commutative diagram) a 1 a 4 f 3 a 1 f 1 a 2 f 2 g 3 g 1 a 3 f 4 a 4 g 2 (2.13) g 2 g 1 = g (groupoid) 2.4 (group) 2.5 Set, Top, rp, Mod, Vct K, Pos (partially ordered set), P (X) 2.6 C D C a, b D F a, F b C f : a b D F f : F a F b F F (g f) = F g F f (2.14) F (1 a ) = 1 F a (2.15) F C D (covariant functor) (functor) F : C D a F a f F f C f a b g f c g F F a F f F b F (g f)=f g F f F g F F c D (2.16) 3
4 C a, b D a, b C f : a b D f : b a (g f) = f g (2.17) (1 a ) = 1 a (2.18) C D (contravariant functor) C f a b g f c g a f b (g f)=f g g c D (2.19) F : C D C a b F D F a F b a b (2.10) f : a b f : b a f f = 1 a, f f = 1 b (2.20) F F (f) : F a F b F (f ) : F b F a (2.14), (2.15) F (f ) F (f) = 1 F a, F (f) F (f ) = 1 F b (2.21) F a F b D 1 a a f F F a F f 1 F a (2.22) C a b f f b 1 b F F a F b D F f 1 F b F F f F b a = b F a = F b (2.23) a = b a = b (2.24) 2.7 P : Set Set, P : Set Set, ϕ : Pos Pos, : Vct K Vct K, U : rp Set, : Set Mod, : Set Vct K, : Set rp 4
5 2.8 C D F, F F F (natural transformation) T : F F C a D T a : F a F a a, b f C a f b F F a F f F b T a T D b F F a F f F b (2.25) Vct Vct Z, Z n, S n, L(n, R), SL(n, R), O(n), SO(n), U(1), U(n), SU(n), R n, E(n) = O(n) R n, 3 U(1) := {z C ; z = 1} = {e iθ ; θ R}, (3.1) T := R/2πZ, SO(2) := {g M(2, R); g g = 1, det g = 1} = {( cos θ sin θ ) } sin θ ; θ R cos θ (3.2) (3.3) 3.2 N K 3.3 Lie 5
6 3.4 X, AutX, Banach 4.3 Euclid Minkowski Hilbert 4.4 Lorentz Sp(n) = {g U(2n); g J n g = J n } Hilbert 6
7 5.2 (intertwining operator, intertwiner) g π π(g) V V T T Vct π V V π (g) (5.1) 2 (π, V ) (π, V ) Hom (V, V ) Hom (V, V ), Schur Schur 2 π, π C Hilbert V, V V, V T : V V g, T π(g) = π (g)t (5.2) (i) π π T = 0. (ii) π π I : V V T = λi λ C π = π T = λ id V (1) Ker T Im T (2) T Ker T V V Im T Hilbert V V Im T V Im T V (3) T 0 (π, V ) Ker T = {0}(π, V ) Im T = V T π π (i) (4) (ii) T I 1 T (π, V ) = (π, V ) T : V V T λ 1 (5.2) T λid V π(g) Ker (T λid V ) V {0} Ker (T λid V ) V (5.3) (π, V ) Ker (T λid V ) = V T = λid V 7
8 (π, V ) (π, V ) T : V V, g, T π(g) = π(g)t λ C, T = λid V (5.4) (outer tensor product representation) π 1 π 2 6 Lie 6.1 Lie Lie Lie Lie Jacobi identity, Lie Lie 6.2 Lie Lie Lie g gl(n, R), gl(n, C), sl(n, R), sl(n, C), o(n), u(n), su(n), o(1, 3) 6.3 R U(1), SU(2) SO(3), SL(2, C) SO (1, 3) 7 Peter-Weyl 7.1 Baire Haar 0 Baire Baire dg = 1 Haar SU(2) Haar p.84, p.64 L(2, R) p.87 8
9 Haar dg (π, V ) v V ṽ := π(g)v dg (7.1) ṽ h π(h)ṽ = π(h) π(g)v dg = π(h)π(g)v dg = π(hg)v dg = π(g)v dg = ṽ (7.2) 7.2 Schur (π, V ), (π, V ) V, V dg Haar v, w V, v, w V v, π(g)w v, π (g)w dg = { 0 (π π ) 1 dim V v, v w, w (π = π ) V, V π(g), π (g) π ij (g), π kl (g) π ij (g)π kl (g) dg = { (1) 0 (π π ) 1 dim V δ ikδ jl (π = π ) (7.3) (7.4) A : V V Hom C (V, V ) (v, v ) A v,v : V V w A v,v (w ) := v v, w (7.5) v 0, v 0 Im A v,v = Cv V 1 (2) Hom C (V, V ) Hom (V, V ), B B := π(g) B π (g 1 ) (7.6) B h Bπ (h) = π(h) B (7.7) B (π, V ) (π, V ) intertwiner 9
10 (3) (7.6) A v,v à v,v (π, V ) (π, V ) intertwiner Schur π = π à v,v = 0 π = π à v,v = λ id V (7.8) λ C (4) (7.3) w, π(g) v v, π (g)w dg = w, à v,v w = { 0 (π π ) λ w, w (π = π ) (7.9) (5) π = π (7.8) Tr à v,v = Tr A v,v = v, v = λ Tr(id V ) = λ dim V (7.10) λ = 1 dim V v, v (7.11) (7.3) 7.3 Haar dg L 2 () := {ψ : C ψ(g) 2 dg < } (7.12) L 2 () Hilbert ϕ, ψ := ϕ(g)ψ(g) dg (7.13) π L (h)ψ(g) := ψ(h 1 g), π R (h)ψ(g) := ψ(gh) (7.14) π L, π R L 2 () π L π R L 2 () ((π L π R )(h l, h r )ψ)(g) := ψ(h 1 l gh r ) (7.15) Ĝ (group dual) (π, V π ) Ĝ d π = dim V π (π, V π ) Ĝ V π f π (g)f := f π(g 1 ) V π (7.16) Φ π : V π V π L 2 () f v Φ π (f, v) : C g Φ π (f, v)(g) := d π f, π(g)v (7.17) 10
11 π π π L π R intertwiner Peter-Weyl Φ : π V π V π L 2 () c = π,i f π i v π i Φ(c) : C g Φ(c)(g) := π,i dπ f i, π(g)v i (7.18) π π π π L π R intertwiner { dπ π ij (g) π Ĝ; i, j = 1, 2,, d π} L 2 () ψ L 2 () (7.19) dπ c π ij π ij (g) (7.20) ψ(g) = π Ĝ i,j c π ij = d π π ij (g)ψ(g) dg (7.21) (7.19) f L 2 (), π Ĝ, i, j 1, 2,, d π, f(g)π ij (g)dg = 0 f 0 (7.22) (1) h(g) h(g 1 ) = h(g) h π ij h 0 (A h ψ)(g) := h(gx 1 )ψ(x) dx (7.23) A h L 2 () 0 Schwarz h ϕ, A h ψ = = ϕ(g)h(gx 1 )ψ(x) dxdg ϕ(g)h(xg 1 )ψ(x) dxdg = A h ϕ, ψ (7.24) (2) A h 0 σ E σ A h p.154 A h ψ E σ π ij 11
12 (3) a, f L 2 () (T a f)(g) := f(ga) (7.25) T a A h T a E σ U a := T a Eσ E σ π ν 1,, ν d ν j (ga) = (U a ν j )(g) = d π ij (a)ν i (g) (7.26) i=1 g = e d ν j (a) = π ij (a)ν i (e) (7.27) i=1 ν j π ij ν j E σ π ij ν j = 0 E σ = {0} E σ {0} h = 0 (4) h(g 1 ) = h(g) f L 2 () π ij f = 0 f L 2 () π ij s(x) = f(xy)f(y) dy (7.28) π ij, s = π ij (x)s(x) dx = π ij (x)f(xy)f(y) dx dy = π ij (zy 1 )f(z)f(y) dz dy = π ik (z)π jk (y)f(z)f(y) dz dy k = 0 (7.29) s π ij (5) h(x) = s(x) + s(x 1 ) h(x) h = 0. s(e) = f(y)f(y) dy (7.30) 0 = h(e) = s(e) + s(e) = 2s(e) f(y) 2 dy = 0. f = 0 12
13 7.4 = π Ĝ(dim V π ) 2, (7.31) Ĝ = (7.32) U(1) SU(2) 8.1 U(1) 8.2 SU(2) SU(2) SU(2) := = { ( α g = β { ( α g = β ) ( ) } γ 1 0 M(2, C) δ g g =, det g = β ) } M(2, C) ᾱ α 2 + β 2 = 1 (8.1) SU(2) = S 3. C 2 ( ) ( ) g SU(2), C 2 z αz + γw v = gv = w βz + δw 2 ψ(z, w) = ψ(v) L SU(2) C 2 (8.2) ψ(v) (U g ψ)(v) := ψ(g 1 v) = ψ(δz γw, βz + αw) (8.3) U g1 U g2 = U g1 g 2 ψ ψ(z, w) = a 0 z n + a 1 z n 1 w + a 2 z n 2 w a n 1 zw n 1 + a n w n (8.4) U g ψ n L n (n = 0, 1, 2, ) SU(2) dim L n = n + 1 L SU(2) ψ(z) dz d z = ψ(x + iy) dx dy = 2π 0 0 ψ(re iθ ) rdr dθ (8.5) 13
14 ϕ, ψ := ϕ(z, w)ψ(z, w) e z 2 w 2 dz d z dw d w (8.6) U g ϕ, U g ψ = ϕ, ψ 2π 0 0 r 2k e r2 rdr dθ = π 0 t k e t dt = π Γ (k + 1) = π k! (8.7) z k w m, z p w q = δ kp δ mq π 2 k! m! (8.8) 2 n L n (n = 0, 1, 2, ) SU(2) n + 1 SU(2) (1) SU(2) ( ) α 0 g 1 = 0 α 1, α C, α = 1, (8.9) ( ) cos θ sin θ g 2 =, θ R (8.10) sin θ cos θ L n A U g1, U g2 ϕ k = z k w n k U g1 ϕ k = α 2k n z k w n k α n, α n 2,, α n α U g1 1 A U g1 A Aϕ k = λ k ϕ k λ k C U g2 z n = (z cos θ + w sin θ) n = A U g2 0 = [A, U g2 ]z n = n z k w n k cos k θ sin n k θ (8.11) k=0 n (λ k λ n )z k w n k cos k θ sin n k θ (8.12) k=0 λ k λ n (k = 0,, n). A = λ n id Schur (L n, U) (2) SU(2) ( ) e iθ 0 g = 0 e iθ, 0 θ π (8.13) SU(2) 2π L n χ n (g) = e inθ + e i(n 2)θ + e i(n 4)θ + + e i( n+2)θ + e inθ (8.14) 14
15 χ n (g) χ n 1 (g) = e inθ + e inθ = cos nθ (8.15) n = 0, 1, 2, 2π SU(2) (Clebsch-ordan ) χ m χ n = χ m+n + χ m+n χ m n (8.16) χ 1 χ 1 = χ 2 + χ = 3 1 χ 2 χ 1 = χ 3 + χ = 5 1 χ 2 χ 2 = χ 4 + χ 2 + χ = χ 5 χ 2 = χ 7 + χ 5 + χ = χ 7 χ 7 = χ 14 + χ χ = (8.17) intertwiner Clebsch-ordan 8.3 SU(3) Young 3 3 = 6 3 (8.18) 3 3 = 8 1 (8.19) 6 3 = 10 8 (8.20) 8 8 = (8.21) p Fourier-Pontryagin duality (locally compact Hausdorff Abelian group) Z p Z U(1) γ γ : U(1) (character) γ : U(1), g γ(g), (9.1) γ(g 1 g 2 ) = γ(g 1 )γ(g 2 ) (9.2) γ(g 1 ) = γ(g) (9.3) 15
16 Ĝ = {γ} (γ 1 γ 2 )(g) := γ 1 (g)γ 2 (g) (9.4) Ĝ (dual group) (character group) Ĝ Ĝ = (9.5) Fourier γ 1 (g)γ 2 (g)dg = δ γ1,γ 2 (9.6) γ(g 1 )γ(g 2 )dγ = δ g1,g 2 (9.7) Ĝ Fourier Hilbert H U U : U(H), g U(g) (9.8) P γ := γ(g)u(g)dg (9.9) (9.10) P γ = γ(g)u(g) dg = γ(g 1 )U(g 1 )dg = γ(g)u(g)dg P γ = P γ, (9.10) P α P β = δ αβ P β, (9.11) P γ dγ = 1, (9.12) Ĝ U(g)P γ = γ(g)p γ, (9.13) P γ U(g) = γ(g)p γ, (9.14) γ(g)p γ dγ = U(g) (9.15) Ĝ = P γ (9.16) 16
17 (9.11) (9.12) P α P β = α(g)u(g)dg β(h)u(h)dh = dg dh α(g)β(h)u(gh) = dg dh α(g)β(g 1 h)u(h) = dh dg α(g)β(g)β(h)u(h) = δ α,β dh β(h)u(h) Ĝ = δ αβ P β (9.17) P γ dγ = = = Ĝ dγ dg Ĝ dg δ g,e U(g) dg γ(g)u(g) dγ γ(g)γ(e) U(g) = U(e) = 1 (9.18) (9.13) U(g)P γ = U(g) γ(h)u(h)dh = γ(h)u(gh)dh = γ(g 1 h)u(h)dh = γ(g 1 )γ(h)u(h)dh = γ(g) γ(h)u(h)dh = γ(g)p γ (9.19) (9.14) (9.15) γ(g)p γ dγ = dγ γ(g) dh γ(h)u(h) Ĝ Ĝ = dh dγ γ(h)γ(g) U(h) Ĝ = dh δ g,h U(h) = U(g) (9.20) 9.2 Doplicher-Roberts category Doplicher-Roberts T 17
18 (i) (ii) 2 π, π Hom(π, π ) T (λ 1 S 1 + λ 2 S 2 ) = λ 1 T S 1 + λ 2 T S 2 (9.21) (λ 1 T 1 + λ 2 T 2 ) S = λ 1 T 1 S + λ 2 T 2 S (9.22) π π π (9.23) λ 1 T S 1 +λ 2 T S 2 S 1 S 2 λ 1 S 1 +λ 2 S 2 π π π T π (iii) Hom(π, π ) Banach (iv) S : π π, T : π π T S T S (v) : T T S : π π S : π π (vi) S S = S 2 (C -norm property) (λ 1 S 1 + λ 2 S 2 ) = λ 1 S 1 + λ 2 S 2 (9.24) (S ) = S (9.25) (T S) = S T (9.26) (vii) E Hom(π, π) E = E, E 2 = E E V Hom(σ, π) V V = 1 σ V E V V = E V : σ π σ E subobject σ 1, σ 2 V 1 : σ 1 π, V 2 : σ 2 π V 1 V1 + V 2 V2 = 1 π π σ 2 V 2 V2 V 1 σ 1 π V1 1 π (9.27) (viii) 2 ρ, τ ρ τ 2 R : ρ ρ, T : τ τ R T : ρ τ ρ τ R, T ρ σ ρ σ (ρ σ) τ ρ (σ τ) (9.28) R S R S (R S) T R (S T ) ρ σ ρ σ (ρ σ ) τ ρ (σ τ ) 18
19 (ix) (R T ) (R T ) = (R R) (T T ) (9.29) ρ τ ρ τ ρ τ ρ τ (9.30) R ρ T τ R R T T R R T T R T ρ τ R T R T ρ τ ρ τ ρ τ ρ τ (x) (R T ) = R T (9.31) (xi) i π i τ = τ i = τ T : τ τ 1 i T = T 1 i = T (9.32) Hom(i, i) = C1 i (9.33) 1 (xii) U : π π U U = 1 π, U U = 1 π τ, ρ ρ τ τ ρ ε(ρ, τ) ε(ρ, τ ) (R T ) = (T R) ε(ρ, τ) (9.34) ε(τ, ρ) ε(ρ, τ) = 1 ρ τ (9.35) ε(i, ρ) = ε(ρ, i) = 1 ρ (9.36) ε(ρ σ, τ) = (ε(ρ, τ) 1 σ ) (1 ρ ε(σ, τ)) (9.37) 19
20 ρ τ ρ τ ρ τ ε(ρ,τ) τ ρ (9.38) R T ε(ρ,τ) R T T R ρ τ τ ρ ρ τ τ ρ ε(ρ,τ ) ρ τ 1 ρ τ (9.39) ε(ρ,τ) τ ρ ε(τ,ρ) ρ τ i ρ ρ ρ i (9.40) ε(i,ρ) 1 ρ ε(ρ,i) ρ i ρ i ρ ρ σ τ ε(ρ σ,τ) ρ τ σ τ ρ σ ε(ρ,τ) 1σ 1 ρ ε(σ,τ) (9.41) (xiii) ρ (conjugate) ρ R : i ρ ρ R := ε( ρ, ρ) R : i ρ ρ ( R 1 ρ ) (1 ρ R) = 1 ρ (9.42) (R 1 ρ ) (1 ρ R) = 1 ρ (9.43) R i R R ρ ρ ρ ρ ε( ρ,ρ) ρ i = ρ 1 ρ ρ ρ ρ i ρ = ρ R 1 ρ 1 ρ R i R R ρ ρ ρ ρ ε(ρ, ρ) i ρ = ρ 1 ρ ρ ρ ρ ρ i = ρ R 1 ρ 1 ρ R (9.44) (9.45) T DR C T (i)-(iv) T Banach (i)-(v) Banach (i)-(vi) C (i), (ii), (viii), (ix) Doplicher-Roberts intertwiner DR Rep() Rep 20
21 DR T T = Rep() π T (9.46) V g V π π V V π T V V π π T π(g) π (g) V V π V π T V Vct intertwiner. DR V Mackey s imprimitivity system Curie intertwiner
22 SU(2) Haar Hopf bundle: SU(2) S 3 = SU(2) U(1) = S 1 ρ S 2 (13.1) SU(2) := {g Mat(2, C) g g = gg = I, det g = 1} = S 3 R 4 (13.2) ( ) y 0 + iy 3 iy 1 + y 2 g = det g = (y 0 ) 2 + (y 1 ) 2 + (y 2 ) 2 + (y 3 ) 2 = 1 (13.3) iy 1 y 2 y 0 iy 3 su(2) R 3 Hopf R 3 = su(2), x = (x1, x 2, x 3 ) x = x 1 σ 1 + x 2 σ 2 + x 3 σ 3 (13.4) ρ : SU(2) = S 3 S 2 su(2), g gσ 3 g = ñ = n 1 σ 1 + n 2 σ 2 + n 3 σ 3 (13.5) fiber ρ 1 (σ 3 ) = {e 1 2 iσ 3ψ ψ R} = U(1) = S 1. Hopf Hopf τ : R 4 R + S 3 R + S 2 R 3, ( r, g) (r, gσ 3 g ) Mat(2, C) q = rg = q 0 + iσ k q k x = qσ 3 q = r gσ 3 g = σ k x k su(2) ( ) ( ) q 0 + iq 3 iq 1 + q 2 x 3 x 1 ix 2 iq 1 q 2 q 0 iq 3 x 1 + ix 2 x 3 q 0 q x 1 2q 0 q 2 + 2q 1 q 3 1 q 2 x 2 = 2q 0 q 1 + 2q 2 q 3 (13.6) x 3 q0 2 q2 1 q2 2 + q2 3 q 3 x 2 = det x = det(qσ 3 q ) = det(σ 3 ) det(q) det(q) = (det q) 2 = q 4 r = q 2 = x (13.7) 22
23 Hopf x 1 q 2 q 3 q 0 q 1 x 2 x 3 = q 1 q 0 q 3 q 2 q 0 q 1 q 2 q 3 0 q 3 q 2 q 1 q 0 q 0 q 1 q 2 q 3 = W q 0 q 1 q 2 q 3 (13.8) W T W = W W T = q 2 I Hopf Euler angles: τ : C 2 R 3, ( ) ψ σ 1 ψ q 0 + iq 3 ψ = = ψ σ 2 ψ (13.9) iq 1 q 2 ψ σ 3 ψ g = e i 2 σ 3ϕ e i 2 σ 2θ e i 2 σ 3ψ x 1 x 2 x 3 (13.10) gσ 3 g = σ 1 sin θ cos ϕ + σ 2 sin θ sin ϕ + σ 3 cos θ (13.11) SU(2) 0 < θ < π, 0 ϕ < 2π, 0 ψ < 4π. g N = e i 2 σ 3ϕ e i 2 σ 2θ e + i 2 σ 3ϕ e i 2 σ 3ψ g S = e i 2 σ 3ϕ e i 2 σ 2θ e i 2 σ 3ϕ e i 2 σ 3ψ Maurer-Cartan 1-form: g dg = i { σ 1 (sin ψ dθ cos ψ sin θ dϕ) 2 } +σ 2 (cos ψ dθ + sin ψ sin θ dϕ) + σ 3 (dψ + cos θ dϕ) = i { } σ1 α 1 + σ 2 α 2 + σ 3 α 3 2 (13.12) (13.13) (13.14) α 1 α 2 α 3 sin ψ cos ψ sin θ 0 dθ = cos ψ sin ψ sin θ 0 dϕ (13.15) 0 cos θ 1 dψ 3-form Trg dg g dg g dg = ( i 2) 3 3! σ1 σ 2 σ 3 α 1 α 2 α 3 = i 6 2i sin θ dθ dϕ dψ 8 = 3 sin θ dθ dϕ dψ 2 (13.16) sin θ dθ dϕ dψ = 2 2π 4π = 16π 2 (13.17) 23
24 Haar SU(2) = S 3 SU(2) Tr(g dg) (g dg) = 1 2 Ω = 1 sin θ dθ dϕ dψ (13.18) 16π2 { dθ 2 + (sin θ dϕ) 2 + (dψ + cos θ dϕ) 2} (13.19) q = rg q = q 0 + iσ k q k Mat(2, C) SU(2) q dq q dq = 1 2 dr + rg dg = η 0 + iσ k η k = q 0 dq 0 + q 1 dq 1 + q 2 dq 2 + q 3 dq 3 +iσ 1 ( q 1 dq 0 + q 0 dq 1 q 3 dq 2 + q 2 dq 3 ) +iσ 2 ( q 2 dq 0 + q 3 dq 1 + q 0 dq 2 q 1 dq 3 ) η 0 η 1 η 2 η 3 = +iσ 3 ( q 3 dq 0 q 2 dq 1 + q 1 dq 2 + q 0 dq 3 ) (13.20) q 0 q 1 q 2 q 3 dq 0 dq 0 q 1 q 0 q 3 q 2 dq 1 q 2 q 3 q 0 q 1 dq 2 = Q dq 1 (13.21) dq 2 q 3 q 2 q 1 q 0 dq 3 dq 3 Q Q T Q = QQ T = q 2 I q 2 = q 2 0 +q2 1 +q2 2 +q2 3 = det q = 1 2 Trq q SU(2) (13.14) (13.20) rg dg (13.21) 1 2 rα k = η k k = 1, 2, 3 (13.22) dq 0 η 0 q 0 q 1 q 2 q 3 dr dq 1 dq 2 = 1 η 1 q 2 QT η 2 = 1 q 1 q 0 q 3 q 2 rα 1 2r q 2 q 3 q 0 q 1 rα 2 dq 3 η 3 q 3 q 2 q 1 q 0 rα 3 q 0 q 1 q 2 q dr = 1 q 1 q 0 q 3 q 2 0 r sin ψ r cos ψ sin θ 0 dθ 2r q 2 q 3 q 0 q 1 0 r cos ψ r sin ψ sin θ 0 dϕ (13.23) q 3 q 2 q 1 q r cos θ r dψ Q = R 4 {0} = R + S 3 g Q := 2 Tr(q dq) (q dq) 1 } = 2{ 2 dr2 + r 2 Tr(g dg) (g dg) = dr 2 + r 2( dθ 2 + (sin θ dϕ) 2 + (dψ + cos θ dϕ) 2) = 4 q 2 (dq dq dq dq 2 3) (13.24) 24
25 SU(2) R 4 X := Q/U(1) = R + S 2 = R 3 {0} g X := 1 2 Trd xd x = dr2 + r 2( dθ 2 + (sin θ dϕ) 2) = dx dx dx 2 3 (13.25) 3 R 3 SU(2) (13.6) τ : Q X g Q Vol Q = (2 q ) 4 dq 0 dq 1 dq 2 dq 3 = dx 1 dx 2 dx 3 rdψ (13.26) dq 0 dq 1 dq 2 dq 3 = 1 16r dx 1 dx 2 dx 3 dψ (13.27) T Q { (T Q) H = span r, 1 r θ, 1 ( r sin θ ϕ cos θ )} ψ { } (T Q) V = span ψ (13.28) (13.29) T Q { } (T Q) H = span dr, r dθ, r sin θ dϕ { } (T Q) V = span r(dψ + cos θ dϕ) (13.30) (13.31) 14 p.64 {( ) } x y = ; x, y, z R, x > 0, z > 0 0 z (14.1) ( ) ( ) ( ) a b x y x y = 0 c 0 z 0 z (14.2) x = ax, y = ay + bz, z = cz (14.3) dx dy dz = a dx (a dy + b dz) c dz = a 2 c dx dy dz = x 2 z x 2 dx dy dz (14.4) z ω L = 1 x 2 z dx dy dz = 1 x 2 dx dy dz (14.5) z 25
26 ( ) ( ) ( ) x y a b x y = 0 z 0 c 0 z (14.6) x = ax, y = bx + cy, z = cz (14.7) dx dy dz = a dx (b dx + c dy) c dz = ac 2 dx dy dz = x z 2 dx dy dz (14.8) xz2 ω R = 1 x z 2 dx dy dz = 1 dx dy dz (14.9) xz2 ω L = xz2 x 2 z ω R = z x ω R (14.10) R gω L = cz ax ω R = c a ω L (14.11) (g) = c a p.84 1 {( ) } x y = ; x, y, z R, x (14.12) (14.13) [1] S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer, S., [2] K. H. Rose, XY-pic L A TEX [3] SC 52, (2006). [4],, (1980) 26
27 [5], (2009). [6] (2001). [7], : (2003). [8] H. (1990), ( ). [9] (1960). [10] (2005). Lie Lie (1,2 ) (1999) [11] (1951). [12] (1994). [13] S. Doplicher and J. E. Roberts, A new duality theory for compact groups, Invent. Math. 98, 157 (1989). [14] pp (2004). [15] No. 10pp , (1998). [16], 2006, (2006). [17] P. W. Anderson, More is different, Science, 177, 393 (1972). [18] 11 3, ( ), 11 4, ( ), 12 1, ( ) web page [19]
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
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