K g g g g; (x, y) [x, y] g Lie algebra [, ] bracket (i) [, ] (ii) x g [x, x] = 0 (iii) ( Jacobi identity) [x, [y, z]] + [y, [z, x]] +

Transcription

1 2015 X V Sophus Lie [Hu] James Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics Volume , Springer [Sa],, 2002, [Hu] [Sa] [Hu] K 0 K R C K Date: January 19,

2 K g g g g; (x, y) [x, y] g Lie algebra [, ] bracket (i) [, ] (ii) x g [x, x] = 0 (iii) ( Jacobi identity) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = g h h g Lie subalgebra (i) h g (ii) x, y h [x, y] h g g dim g 2.3. V [x, y] := 0 (x, y V ) V [g, g] = 0 g f : g 1 g 2 f homomorphism (i) f (ii) x, y g 1 f([x, y]) = [f(x), f(y)] f isomorphism g 1, g 2 g 1 g 2 g 1 = g n g 2 g [x, y] = y g {x, y}. g e 1, e 2 [e 1, e 2 ] = ae 1 + be 2 a, b K g (a, b) (0, 0) b 0 [ 1 b e 1, [e 1, e 2 ]] = [ 1 b e 1, ae 1 + be 2 ] = [e 1, e 2 ]. x := 1 b e 1, y := [e 1, e 2 ] [x, y] = y {x, y}

3 2015 X V g 1, g [x i, y i ] = y i g i {x i, y i } i = 1, 2 f : g 1 g 2 f(x 1 ) = x 2, f(y 1 ) = y 2 f 2 g = ( ) 0 1, 0 0 ( ) 1 0 M (K) K [x, y] := xy yx g Sophus Lie 3 3 Luigi Bianchi R 3 e 1, e 2, e 3 x = x 1 e 1 + x 2 e 2 + x 3 e 3, y = y 1 e 1 + y 2 e 2 + y 3 e 3 R 3 x 1 x 2 x 3 [x, y] := x y = y 1 y 2 y 3 e 1 e 2 e 3 = (x 2y 3 y 2 x 3, x 3 y 1 y 3 x 1, x 1 y 2 y 1 x 2 ) R (Heisenberg ). g 3 g {x, y, z} g 3 Heisenberg [x, y] = z, [x, z] = 0, [y, z] = 0. 3 Heisenberg 3 x = , y = , z = M 3 (K) g v, w g [v, w] := vw wv g 3 Heisenberg , Heisenberg n V End(V ) := {f : V V } [x, y] := xy yx (x, y End(V )) End(V ) End(V ) gl(v ) gl n (K) general Lie algebra gl n (K)

4 4 K n M n (K) gl n (K) := M n (K), [x, y] := xy yx (x, y M n (K)) gl(v ) gl(v ) linear Lie algebra 2.15 (A l ). l + 1 V sl(v ) := sl l+1 (K) := {x gl(v ) Tr(x) = 0} special Lie algebratr(x) x sl(v ) dim sl(v ) = (l + 1) 2 1 (i, j) 1 0 (l + 1) e ij h k := e kk e k+1,k+1 {e ij, h k i j, 1 k l} sl(v ). Tr(x) = 0 V sl(v ) well-defined x, y gl(v ), a, b K Tr(ax + by) = atr(x) + btr(y), Tr(xy) = Tr(yx) sl(v ) Tr : gl(v ) K K dim sl(v ) = dim gl(v ) 1 = (l + 1) 2 1 {e ij, h k i j, 1 k l} 2.17 (C l ). 2l V V ( ) V O Il M = V I l O ω : V V K; (v, w) t vmw v, w v = (v i ), w = (w i ) l l ω(v, w) = v i w l+i w i v l+i ω sp(v ) := sp 2l (K) := {x gl(v ) ω(x(v), w) = ω(v, x(w)), v, w V } symplectic algebra ω(x(v), w) = ω(v, x(w))

5 2015 X V 5 t xm + Mx = 0 sp 2l (K) = {x M 2l (K) t xm + Mx = 0} sp(v ) dim sp(v ) = 2l 2 + l sp(v ) dim sp(v ) = 2l 2 + l (i, j) 1 0 2l e ij sp(v ) e ii e l+i,l+i (1 i l), e ij e l+j,l+i (1 i j l) e i,l+j + e j,l+i, e l+i,j + e l+j,i (1 i < j l), e i,l+1i, e l+1,i (1 i l) 2.19 (B l ). 2l + 1 V V M = O I l V 0 I l O ω : V V K; (v, w) t vmw v, w v = (v i ), w = (w i ) l+1 l+1 ω(v, w) = v 1 w 1 + v i w l+i + v l+i w i i=2 ω o(v ) := o 2l+1 (K) := {x gl(v ) ω(x(v), w) = ω(v, x(w)), v, w V } orthogonal Lie algebra ( ) O Il 2.20 (D l (l 2)). 2l V M = I l O V i=2 ω : V V K; (v, w) t vmw o(v ) := o 2l (K) := {x gl(v ) ω(x(v), w) = ω(v, x(w)), v, w V } orthogonal Lie algebra o 2l+1 (K) o 2l (K) dim o 2l+1 (K) = 2l 2 + l, dim o 2l (K) = 2l 2 l A l, B l, C l, D l (i) t n (K) := { (a ij ) gl n (K) a ij = 0 if i > j }

6 6 (ii) n n (K) := { (a ij ) gl n (K) a ij = 0 if i j } (iii) d n (K) := { (a ij ) gl n (K) a ij = 0 if i j } t n (K), n n (K), d n (K) (i) t n (K) = d n (K) + n n (K) (ii) [d n (K), n n (K)] = n n (K) (iii) [t n (K), t n (K)] = n n (K) t n (K), n n (K), d n (K) (Ado (1936)). g 2.32 g K V f : V V V ; (x, y) xy K V f K K V Der(V ) := {D gl(v ) D(xy) = xd(y) + D(x)y, x, y V } V derivation algebra Der(V ) Der(V ) K[x] Der(K[x]) = { a(x) d dx } a(x) K[x] K[x] 2.27 Der(K[x]). D Der(K[x]) D(1) = D(1 2 ) = D(1)1 + 1D(1) = 2D(1) D(1) = 0 a(x) := D(x) D(x i ) = ix i 1 a(x) (i Z 1 ) f(x) = n i=0 a ix i K[x] n n D(f(x)) = a i D(x i ) = a i ix i 1 d a(x) = a(x) dx (f(x)) a(x) K[x] a(x) d dx Der(K[x])

7 2015 X V g K ad : g gl(g); x (y ad(x)(y) := [x, y]) Im(ad) Der(g). ad(x) gl(g) ad welldefined ad(x)([y, z]) = [ad(x)(y), z] + [y, ad(x)(z)] g 2.30 ad : g gl(g) g adjoint representation Der(g) Im(ad) inner derivationouter derivation g f : g gl(v ) (representation) V a g x g, y a [x, y] a a g (ideal) 3.2. g a, b g (i) 0 g (ii) Z(g) := { z [x, z] = 0, x g } (center) { m } (iii) [a, b] := [x i, y i ] x i a, y i b, m 1 { m } (iv) Dg := [g, g] := [x i, y i ] x i, y i g, m 1 (derived ideal) g = gl n (K) Dg = sl n (K). x, y gl n (K) Tr([x, y]) = 0 Dg sl n (K) n gl n (K) e i,j e ij 2.16 e ij (i j), h i := e ii e i+1,i+1 (1 i n 1) sl n (K) 3.5 e ij = [e ij, e jj ], h i = [e i,i+1, e i+1,i ] Dg sl n (K)

8 [e ij, e kl ] = δ jk e il δ li e kj g x g δ Der(g) [δ, ad x] = ad(δx) ad(g) Der(g) ad(g) Der(g) x, y g δ Der(g) [δ, ad x](y) = (δ ad x)(y) (ad xδ)(y) = δ[x, y] [x, δy] = [δx, y] + [x, δy] [x, δy] = ad(δx)(y) 3.7. g (simple) (i) g 0 g (ii) Dg (i) (i) (ii) dim g = 1 Dg = 0 g g 1 g (i) dim g = 1 (ii) dim g = g Z(g) = 0 Dg = g g = sl 2 (K). g x = ( ) 0 1, y = 0 0 ( ) ( ) , h = [x, y] = h, [h, x] = 2x, [h, y] = 2y a 0 g 0 ax + by + ch a ad(x) 2 (ax + by + ch) = ad(x)(bh 2cx) = 2bx, ad(y) 2 (ax + by + ch) = ad(y)( ah + 2cy) = 2ay 2bx, 2ay a a 0 y a h = [x, y] a, 2x = [h, x] a a = g b 0 a = g a = b = 0 ch a h a a = g 3.7 (i) dim g = (ii)

9 2015 X V sl n (K) (n 2)sp n (K), so n (K) g a g/a x, y g/a [x, y] := [x, y] g/a g/a quotient (residue) algebra g/a well-defined g/a f : g 1 g 2 (i) Kerf g 1 (ii) Imf g 2 (iii) [ ] g 1 /Kerf = Imf ( 2.25 ). g. g ad : g gl(g) Ker(ad) = Z(g) = g g D 0 g := g, D i g := [D i 1 g, D i 1 g] (i > 0) g = D 0 g D 1 g D 2 g D i g (derived series) n D n g = 0 g (solvable) t n (K) n n (K) = Dt n (K) 2.23 e ij (i j) t n (K) [e ii, e ij ] = e i,j (i < j) n n (K) Dt n (K) t n (K) = d n (K)+n n (K) d n (K) n n (K) = Dt n (K)

10 10 e ij (i < j) j i e ij p e ij t n (K) g p 0 0 x 1,p+1 x 1,n. { }.. g p = x n p,n x ij K g 0 = t n (K), g 1 = n n (K) p Dg p g 2p p = j i = m l n i, j, l, m i m [e ij, e lm ] = δ jl e im δ mi e lj = δ jl e im i m j = l e im m i = (m l) + (l i) = (m l) + (j i) = 2p, j l 0 i = m j > i = m > l [e ij, e lm ] = δ jl e im δ mi e lj = e lj j l = (j i) + (i l) = (j i) + (m l) = 2p [e ij, e lm ] g 2p Dg p g 2p D i t n (K) = D i 1 g 1 = D i 2 g 2 = D i 3 g 2 2 = = g 2 i 1 2 i 1 > n 1 D i t n (K) = g (i) g g (ii) a g g/a g (iii) a, b g a + b. (i) h D i h D i g g h f : g 1 g 2 f(d i g 1 ) = D i g 2 (ii) g/a n D n g a D n+1 g = D 1 (D n g) = [D n g, D n g] D 1 a D n+i g D i a g (iii) a/a b = a + b/b a a/a b a + b/b a (i) a + b/b b (ii) g 1 rad(g) g (radical)

11 2015 X V 11. g a g 3.20 b a + b a a + b = a b a rad(g) = 0 g (semisimple) g g = sl 2 (K) ( ) ( ) y =, h = g 3.25 ( ). g. g ad : g gl(g) Ker ad = Z(g) DZ(g) = 0 Z(g) g g C 0 g := g, C i g := [g, C i 1 g] (i > 0) g = C 0 g C 1 g C 2 g C i g (descending series, lower central series) n C n g = 0 g (nilpotent) g D i g C i g g = t n (K) C 1 t n (K) = D 1 t n (K) = n n (K) C 2 t n (K) = [t n (K), n n (K)] = n n (K) i C i t n (K) = n n (K) g = t n (K) g. 2.7 [x, y] = y g {x, y} D 1 g = C 1 g = y K, D 2 g = 0 C 2 g = [ x, y K, y K ] = y K = C 1 g

12 K 0 g = sl 2 (K) 3.10 Dg = g K 2 g = sl 2 (K) 3.10 g x = ( ), y = ( ), h = ( ) [x, y] = h, [h, x] = 2x = 0, [h, y] = 2y = 0 C 1 g = h K C 2 g = g (i) g g (ii) g/z(g) g (iii) g g 0 Z(g) g x i, y g ad x 1 ad x 2 ad x n (y) = 0 x g (ad x) n = g x ad x 3.32 ( (Engel s Theorem)). g x ad x g x gl(v ) ad x. n > 0 x n = 0 gl(v ) λ x := (y x y), ρ x := (y y x) gl(gl(v )) ad x = λ x + ρ x (λ x ρ x )(y) = λ x ( y x) = x y x = ρ x (λ x (y)) = (ρ x λ x )(y) λ x ρ x 2n 1 ( ) 2n 1 ad x 2n 1 = λ i x ρ 2n 1 i x i i=0 0 i 2n 1 2n 1 i n i n ad x 2n 1 = x gl(v ) ad x x x x ad x = 0

13 2015 X V g gl(v ) V 0 g.v = 0 v V \ {0}. n V gl(v ) n dim g dim g = 0 g = 0 v V \ {0} dim g = 1 g O A g = A K A n = O n A 0 v dim g 2 g h g h g 3.33 h h h g/h h g g/h x h [h, x] h x g h N g (h) := {x g [x, h] h} h g g = N g (h) h g g/h dim g/h > 1 g/h 1 p g h h dim g/h = 1 dim h = dim g 1 z g \ h g = h + Kz W := {v V h.v = 0} W 0 g = h + Kz W z h g W g x g, y h, w W y(xw) = yx(w) = xy(w) [x, y](w) = 0 W z ( ). dim g dim g = 0, 1 dim g > 1 ad g gl(g) 3.35 [g, x] = 0 x g \ {0} Z(g) 0 dim g/z(g) < dim g g/z(g) 3.30 (ii) g n V 0 = V 0 V 1 V n = V (dim V i = i) V (flag) g.v i V i 1 ( i) g V (0 = V 0 V 1 V n = V ) V g n n (K)

14 v V V 1 := v K W = V/V 1 g dim V g W V V g a 0 a Z(g) 0 Z(g) 0. g a 3.35 [g, x] = 0 x a K K g gl(v ) V 0 g dim g dim g = 0 dim g > 0 Step 1 1 a g g D 1 g g g/d 1 g g/d 1 g 1 a Step 2 x.v = λ(x)v ( x a) v V λ : a K x a x.v = λ(x)v v V \ {0} λ(x) K λ Step 3 W := {w V x.w = λ(x)w ( x a)} g Step 4 Step 3 a g 1 g = a + Kz Step 3 z W K z K α z v 0 W x a c K (x + cz).v 0 = x.v 0 + cz.v 0 = λ(x)v 0 + cαv 0 = (λ(x) + cα)v 0 v 0 Step 3 w W, x g, y a yx.w = xy.w [x, y].w = λ(y)x.w λ([x, y])w

15 2015 X V 15 λ([x, y]) = 0 w W, x g w, x.w,, x n.w n W 0 := 0, W i =: w, x.w,, x i 1.w K dim W n = n, W n = W n+i (i 0) x(w n ) W n x W n y a y(w i ) W i W n w, x.w,, x n 1.w y a λ(y) yx i.w λ(y)x i.w (mod W i ) i = 0 i > 0 yx i.w = yxx i 1.w = xyx i 1.w [x, y]x i 1.w xyx i 1.w (mod W i 1 ) yx i 1.w λ(y)x i 1.w (mod W i 1 ) W i x(w i 1 ) W i y a Tr Wn (y) = nλ(y) [x, y] a Tr Wn ([x, y]) = nλ([x, y]) x(w n ) W n, y(w n ) W n [x, y] = xy yx W n 2 x, y Tr Wn ([x, y]) = 0 K 0 λ([x, y]) = K 4.1 K p > 0 p x = , y = diag(0, 1, 2,..., p 1) [x, y] = x g := x, y K gl p (K) g 2 y K p e 1, e 2,, e p x 4.3 ( (Lie s Theorem)). K g gl(v ) V V g t n (K) K g dim a i = i 0 = a 0 a 1 a n = g. g ad : g gl(g) ad(g) 4.3 ad(g) 4.5. K g x Dg ad x Dg g g Dg

16 ad g t n (K) ad Dg = D ad g Dt n (K) = n n (K) K K = C pp A M n (K) Ax = αx x A (eigenvector) α (eigenvalue), V α := Ker(A αi) α A eigenspace f A (x) := xi A A (characteristic polynomial) 4.6. A M n (K) α A A f A (x). α A Ker(A αi) 0 A αi (Hamilton-Cayley ). A M n (K) f A (x) f A (A) = O 4.8. A M n (K) f(a) = O f(x) K[x] A (minimal polynomial) φ A (x) 4.9. A M n (K) A φ A (x) f(a) = 0 f(x) K[x] φ A (x). 4.7 M n (K) K n 2 I, A, A 2,, A n2 K φ A (x) ϕ A (x) A (φ A ϕ A )(x) (φ A ϕ A )(A) = O deg(φ A ϕ A )(x) < deg(φ A (x)), deg(ϕ A (x)) (φ A ϕ A )(x) φ A f(a) = 0 f(x) K[x] f(x) = p(x)φ A (x)+q(x) deg q(x) < deg φ A (x)q(a) = O q = I M n (K) f I (x) = (x 1) n φ I (x) = x A M n (K) I(A) := {f(x) K[x] f(a) = O} I(A) K[x] K[x]

17 2015 X V 17 I(A) = (φ A (x)) φ A (x) K[x] A A M n (K) A φ A (x) f A (x). 4.9 φ A (x) f A (x) φ A (x) α x φ A (α)x = φ A (A)x = 0 x 0 φ A (α) = 0 φ A (x) 4.13 ( P ). A M n (K) (i) A s (ii) φ A (x) φ A (x) = (x α i ) (iii) K n A K n = s V αi.. (i) (ii) A P P 1 AP φ A (x) = φ P 1 AP (x) A A α 1,, α s (A α 1 I)(A α 2 I) (A α s I) = O s φ(x) = (x α i ) φ(a) = O φ(x) φ(x) = φ A (x) (ii) (iii) φ A (x) s α 1,, α s 4.12φ A (x) = (x α i ) (A α 1 I)(A α 2 I) (A α s I) = O A, B M n (K) rank(ab) ranka + rankb n rank(a α 1 I)(A α 2 I) rank(a α 1 I) + rank(a α 2 I) n 2 2 rank{ (A α i I)}(A α 3 I) rank{ (A α i I)} + rank(a α 3 I) n 3 rank(a α i I) 2n

18 18 rank s (A α i I) dim V αi s rank(a α i I) (s 1)n s (n rank(a α i I)) n. = n rank(a α i I) s V αi K n V = (iii) (i) K n = s dim V αi n s V αi. s V αi V αi K n V x End(V ) 4.13 x (semisimple) ( ) A = f 1 0 A (x) = x = (x + 1)(x 1) A R f A (x) K V x End(V ) Λ V x(λ) Λ Λ x V Λ V V = Λ Λ Λ Λ V x End(V ) x x x. x 4.13 V x V = V α x Λ V Λ α := Λ V α x Vα = αi V α Λ α Λ α Λ := Λ α Λ Λ x x x z End(Λ) z W Λ Λ z V = Λ Λ x := z 1 Λ End(V ) W Λ z W V x x W V Λ = W (Λ W )

19 2015 X V 19 x V α xx V α x V α End(V α) V β x V α V β V α = V β V β V 4.13 x V x End(V ) x 0 V x V x End(V ) Λ V x x Λ Λ x Λ x = x Λ x Λ x, x Λ, x Λ A, B, C ( ) B O A = O C A φ A (x) φ A (B) = O, φ A (C) = O x Λ x Λ φ A A, B M n (K) AB = BA A B P P 1 AP P 1 BP A ± B. A α 1,, α s A K n V αi s K n = V αi x V αi A(Bx) = B(Ax) = B(α i x) = α i (Bx) Bx V αi V αi B α 1 I n1 O B 1 O A..., B... O α s I ns O X, Y P Y = P 1 XP X Y B i B V αi 4.20 B i V αi B f 1 (x),, f s (x) K[x] h 1 (x)f 1 (x) + + h s (x)f s (x) = 1 h 1 (x),, h s (x) K[x] B s

20 ( ). A M n (K) s f A (t) = (t α i ) m i α i Ṽα i. f A (t) Ṽ αi := {v V (A α i I) m i v = 0} f i (t) = V = f A(t) (t α i ) m i s Ṽ αi = j i(t α j ) m j f 1 (t),, f s (t) 4.22 (1) h 1 (t)f 1 (t) + + h s (t)f s (t) = 1 h 1 (t),, h s (t) K[t] A i := h i (A)f i (A) A A s = I i j f i (t)f j (t) f A (t) Hamilton- Cayley f i (A)f j (A) = O A i A j = O (i j) A i = A i E = A i (A A s ) = A 2 i A 2 i = A i A 1,, A s s V = A i (V ), A i (V ) = {v V A i v = v} A i (V ) Ṽα i (t α i ) m i f i (t) = f A (t) (A α i I) m i f i (A) = O (A α i I) m i A i = (A α i I) m i h i (A)f i (A) = h i (A)(A α i I) m i f i (A) = O A i (V ) Ṽα i v Ṽα i v A i (V ) (A α i I) m i v = 0 (t α i ) m i h i (t)f i (t) h i (t)f i (t) t α i (1) 4.22 g 1 (t), g 2 (t) K[t] g 1 (t)(t α i ) l +g 2 (t)h i (t)f i (t) = 1 g 1 (A)(A α i I) l +g 2 (A)A i = I

21 2015 X V 21 v v = g 2 (A)A i v = A i (g 2 (A)v) A i (V ) s V = Ṽ αi 4.25 (). s α i m i s f(t) = (t α i ) m i K[t] K[t]/(f(t)) = s K[t]/((t α i ) m i ) 4.27 (Jordan-Chevalley ). V x End(V ) (i) x s, x n End(V ) x = x s + x n, x s x n [x s, x n ] = 0. (ii) x s = p(x), x n = q(x)p(t), q(t) K[t] x s, x n x (iii) A B V x(b) A x s (B), x n (B) A x = x s + x n x Jordan-Chevalley (Jordan ) x s x (semisimple part), x n x (nilpotent part). (iii) (ii) (i), (ii) x f x (t) s f x (t) = (t α i ) m i 4.24 s V = 0 x f(t) := f x (t) K[t]f(t) := tf x (t) K[t] p(t) K[t] p(t) α i mod (t α i ) m i, p(t) 0 mod t q(t) := t p(t), x s := p(x), x n := q(x) Ṽ αi x = x s + x n, [x s, x n ] = 0, p(0) = q(0) = 0

22 22 x s Ṽαi = α i IṼαi x s = α i IṼαi x s x n Ṽαi = (x x s ) Ṽαi = x α i IṼαi (x n Ṽαi ) m i = 0 m := max{m i } x m n = 0 x n x s, x n s n x = s + n [s, n] = 0 [s, x] = 0, [n, x] = 0 [x s, s] = [p(x), s] = 0 [x n, n] = 0 x s s x n n 4.21 x s s x n n x = x s + x n = s + n x s s = n x n x s = s, x n = n x End(V ) x = x s + x n ad x = ad x s + ad x n ad x. [ad x s, ad x n ] = ad[x s, x n ] = (i) ad x s ad x n 3.33 ad x s V x s diag(a 1,, a n ) gl(v ) {e i,j } ad x s (e i,j ) = [x s, e i,j ] = (a i a j )e i,j ad x s K V δ Der(V ) End(V ) End(V ) δ δ = σ + ν σ, ν Der(V ). σ Der(V ) δ V V = l Ṽα i Ṽα i := {v V (δ α i I) m v = 0 (m 0)} α i δ σ x Ṽα i, y Ṽα j n ( ) n (δ (α i + α j )I) n (xy) = ((δ α i I) n k x) ((δ α j I) k y) k k=0 n Ṽα i Ṽα j Ṽ αi +α j σ(xy) = (α i + α j )xy (σx)y + x(σy) = (α i + α j )xy V Ṽα i σ Der(V ) g Dg g 3.32 x Dg ad Dg x Dg x Dg ad Dg x g

23 2015 X V A B gl(v ) M := {x gl(v ) [x, B] A} x M y M Tr(xy) = 0 x. x M x = s + n (s = x s, n = x n ) V s = diag(a 1,, a m ) Q E := a 1,, a m Q K s = 0 E = 0 E E 0 f : E Q f = 0 f E y := diag(f(a 1 ),, f(a m )) gl(v ) {e i,j } ad s(e i,j ) = (a i a j )e i,j, ad y(e i,j ) = (f(a i ) f(a j ))e i,j f f(a i ) f(a j ) = f(a i a j ) r(t ) K[T ] r(a i a j ) = f(a i a j ) ( i, j) i, j ad y(e i,j ) = (f(a i ) f(a j ))e i,j = r(a i a j )e i,j = r(ad s)(e i,j ) ad y = r(ad s) 4.28 ad s = (ad x) s 4.27 p(t ) K[T ] ad s = p(ad x) ad y = r(ad s) = (r p)(ad x) ad x(b) A ad y(b) A y M Tr(xy) = 0 a i f(a i ) = 0 E f f(a i ) 2 = 0 f(a i ) Q i f(a i ) = 0 f = x, y, z End(V ) Tr([x, y]z) = Tr(x[y, z]). Tr([x, y]z) = Tr((xy yx)z) = Tr(xyz xzy) = Tr(x(yz zy)) = Tr(x[y, z]) 4.32 ( ). g gl(v ) x Dg y g Tr(xy) = 0 g x Dg A = Dg, B = g M = {x gl(v ) [x, g] Dg} g M x Dg y g Tr(xy) = x Dg y M Tr(xy) = 0 g x i Dg [x i, x j ] y M 4.31 Tr([x i, x j ]y) = Tr(x i [x j, y]) = Tr([x j, y]x i ) M [x j, y] Dg g x Dg y g Tr(ad x ad y) = 0 g

24 24. gad g Ker ad = Z(g) g 5. B : g g K; (x, y) Tr(ad x ad y) g (Killing form) B g B([x, y], z) = B(x, [y, z]) 5.2. B([x, y], z) = B(x, [y, z]) 5.3. g a B B a B a = B a a. V W V ϕ ϕ(v ) W Trϕ = Tr(ϕ W ) x, y a ad x ad y gl(g) (ad x ad y)(gl(g)) a B(x, y) = Tr(ad x ad y) = Tr((ad x ad y) a ) = Tr(ad a x ad a y) = B a (x, y) 5.4. g a a := {x g B(x, y) = 0 ( y a)} g B (radical) g = 0 B (nondegenerate) 5.5. sl {x, h, y} ad h = diag(2, 0, 2), ad x = , ad y = B B g B (i) g = gl n (K) B(x, y) = 2nTr(xy) 2Tr(x)Tr(y). (ii) g = sl n (K) B(x, y) = 2nTr(xy). (iii) g = so n (K) B(x, y) = (n 2)Tr(xy).

25 2015 X V 25 (iv) g = sp 2n (K) B(x, y) = (2n + 2)Tr(xy).. g = gl n (K) gl n (K) e ij (ad(e ij ) ad(e jl ))(e gh ) = ad(e ij )(δ lg e kh δ hk e gl ) = δ lg [e ij, e kh ] δ hk [e ij, e gl ] = δ lg (δ jk e ih δ hi e kj ) δ hk (δ jg e il δ li e gj ) = δ lg δ jk e ih δ lg δ hi e kj δ hk δ jg e il + δ hk δ li e gj gl n (K) (e ij, e kl ) := δ ik δ jl (ad(e ij ) ad(e jl ))(e gh ) gl n (k) e gh (ad(e ij ) ad(e jl ))(e gh ) gl n (k) e gh (δ lg δ jk e ih δ lg δ hi e kj δ hk δ jg e il + δ hk δ li e gj, e gh ) = δ lg δ jk δ ig δ lg δ hi δ kg δ jh δ hk δ jg δ ig δ lh + δ hk δ li δ jh Tr(ad(e ij ) ad(e jl )) = g,h (δ lg δ jk δ ig δ lg δ hi δ kg δ jh δ hk δ jg δ ig δ lh + δ hk δ li δ jh ) = nδ li δ jk δ kl δ ij δ ij δ lk + nδ kj δ li = 2nδ li δ jk 2δ kl δ ij = 2nTr(e ij e kl ) 2Tr(e ij )Tr(e kl ) B g = gl n (K) B(x, y) = 2nTr(xy) 2Tr(x)Tr(y) sl n (K) gl n (K) 5.3 B(x, y) = 2nTr(xy) g = so n (K) g = sp 2n (K) 5.7. g = so n (K) g = sp 2n (K) g a, b (i) a g (ii) (a + b) = a b (iii) a (a ) (iv) g B det(b(e i, e j )) 0 {e i } g 5.9. K (i) g (ii) Dg g. Dg g (2) Tr([ad x, ad y] ad z) = 0 ( x, y, z g)

26 26 (2) 4.33 g g 4.5 D(ad g) n n (K) ad g t n (K) (2) K g (i) g (ii) g 0. a a rad(g) g rad(g) = 0 a = 0 rad(g) 0 D k rad(g) 0 D k+1 rad(g) = 0 k D k rad(g) g g B g rad(g). x g y D(g ) Tr(ad x ad y) = 0 ad(g ) ad(g ) = g /Z(g ) g g rad(g) g (i) g (ii) g B (i) (ii) (ii) (i) 5.11 g a g x a, y g ad x ad y : g g a (ad x ad y) 2 (g) Da = 0(ad x ad y) 2 B(x, y) = Tr(ad x ad y) = 0 a g = sl n (K), so n (K), sp 2n (K) g a (a ) = a g a i (1 i n) g = n a i g a 1,, a n (direct sum) i j [a i, a j ] a i a j = g a g = a a a. a g a a g B a a 0 a a g a a = 0 dim a+dim a = dim g

27 2015 X V 27 {e 1,, e m } a g {e 1,, e n } B := (B(e i, e j )) = (b 1,, b n ) B n b i K n n { n } a = a i e i g B(x, e i ) = 0, 1 i m = = { n } a i e i g (a 1,, a n )b i = 0, 1 i m { n } a i e i g B t (a 1,, a n ) = 0 B := t (b 1,, b m ) B B : K n K m a = Ker B dim a = dim Ker B = n rank B = dim g dim a a a 5.13 B a a a \ {0} B a (a, x) = 0 ( x a) 5.3 B(a, x) = 0 ( x a) a a a = {0} g a b a b g. b g [b, g] b 5.17 g = a a [b, g] = [b, a a ] = [b, a] [b, a ] [b, a ] [a, a ] a a = 0 [b, g] = [b, a] b g 1 G n A A G g, a a g a a b a b g G A A a G b G n = 2 { g := 0 θ x } { θ 0 y a := 0 0 x } { 0 0 y b := 0 0 x } O

28 g g g i (1 i n) g = n g i a g a g i B gi = B gi g i. g Dg 0 g g g g g g = g 1 g 1 g 1, g g 1 g 1 g = n g i a g [a, g] = 0 a Z(g) rad(g) = 0 [a, g] a g [a, g] = a [a, g] = [a, g 1 ] [a, g n ] 0 a = [a, g i ] g i i g i a = g i g g = Dg g g g g i (1 i n) g = n g i [g i, g j ] = 0 (i j) Dg i = g i Dg = [ n g i, n g i ] = n g i = g ϕ : g h h 5.20 g h g ad g = Der(g). g Z(g) = 0 g = ad g h := ad g, d := Der(g) 3.6 h d 5.3 B h d B d B d h := {x d B d (x, y) = 0, y h} B h (h ) h := {x h B h (x, y) = 0, y h} h h = (h ) h g h h = (h ) h = 0 h h d [h, h] h h = δ h d ad(δx) = [δ, ad x] = 0 ( x g) g = ad g δx = 0 ( x g) δ = 0 h = 0 {e 1,, e m } h d {e 1,, e n } 5.17 h = Ker B B := (B d (e i, e j )) = (b 1,, b n ) B := t (b 1,, b m ) h = 0 Ker B = 0 m = n Der(g) = ad g

29 2015 X V (). g 5.22 ad : g = ad g = Der(g) x g ad x Der(g) End(g) ad x = (ad x) s +(ad x) n (ad x) s, (ad x) n Der(g) 4.29 ad : g = Der(g) ad(x s ) = (ad x) s, ad(x n ) = (ad x) n x s, x n g x = x s + x n, [x s, x n ] = 0 x g ( ) (Jordan decomposition x s x (semisimple part) x n x (nilpotent part) g ( 5.13, 5.17, 5.20 ). g (i) g (ii) g 0 (iii) g B (iv) g a a a = {0} (v) g g i (1 i n) g = n g i 6.1. g g V V g V V ; (x, v) xv 3 g (g-module) (M1) (ax + by).v = a(x.v) + b(y.v), (M2) x.(av + bw) = a(x.v) + b(x.w), (M3) [x, y].v = x.y.v y.x.v (x, y g; v, w V ; a, b K) g V g ϕ : g gl(v ) V g g gl(v ); x (v x.v) g ϕ : g gl(v ) x.v := ϕ(x)(v) V g 6.3. g V, W ϕ(x.v) = x.ϕ(v) ϕ : V W g (homomorphism) g (isomorphism) 6.4. V g V W g.w W W g g V 0 V g V irreducible ( ) V g V (completely reducible) 6.5. g ϕ : g gl(v )

30 30 (i) V (ii) V g W W V = W W. (i) (ii) V = V i (V i ) W V W W W = 0 V {0} V W V = W + W W + W V i 0 V i0 W + W V i0 (W + W ) V i0 V i0 V i0 (W + W ) = 0 W + V i0 (W + V i0 ) W = 0 W W + V i0 W (ii) (i) V W V = W W V (i) V = W W W W 0 U U W W U W 6.6 (Schur ). ϕ : g gl(v ) f gl(v ) ϕ(x) (x g) f. λ f f λ 1 V gl(v ) V λ := Ker(f λ 1 V ) v λ x g f(ϕ(x)(v)) = ϕ(x)(f(v)) = ϕ(x)(λv) = λ(ϕ(x)(v)) V λ g ϕ V λ = V f = λ 1 V 6.7. g V, W g (i) V := Hom K (V, K) V g V V ; (x, f) x.f := [v f(x.v)] V g (ii) V, W K V K W g (V K W ) V K W ; (x, v w) x.v w + v x.w V K W g (iii) V W Hom K (V, W ) g Hom K (V, W ) Hom K (V, W ); (x, f) x.f := [v x.f(v) f(x.v)] Hom K (V, W ) g. (i) 6.1 (M1), (M2) (M3) [x, y].f(v) = f([x, y].v) = f(x.y.v y.x.v) = f(x.y.v) + f(y.x.v) = (x.y.f)(v) (y.x.f)(v) = ((x.y y.x).f)(v).

31 2015 X V 31 (ii) (M3) [x, y].(v w) = ([x, y].v) w + v ([x, y].w) (iii) = (x.y.v y.x.v) w + v (x.y.w y.x.w) = (x.y.v w + v x.y.w) (y.x.v w + v y.x.w)) = (x.y y.x).(v w) g β S := {x g β(x, y) = 0 ( y g)} (radical) S = {0} β (nondegenerate) β([x, y], z) = β(x, [y, z]) ( x, y, x g) (associative) 6.9. g ϕ : g gl(v ) β(x, y) := Tr(ϕ(x)ϕ(y)) β g. β 4.31 β S β S g ϕ(s) = S S g S = 0 β g ϕ : g gl(v ) β(x, y) := Tr(ϕ(x)ϕ(y)) g {x 1,, x n } β {y 1,, y n } g g β β g = g {y 1,, y n } g c ϕ (β) := n ϕ(x i )ϕ(y i ) End(V ) ϕ (Casimir operator) β c ϕ c ϕ x g ϕ(x) c ϕ : V V g Tr(c ϕ ) = dim g ϕ c ϕ (dim g/ dim V ) g. [x, x i ] = j a ijx j, [x, y i ] = j b ijy j a ik = a ij β(x j, y k ) = β([x, x i ], y k ) = β(x i, [x, y k ]) = β(x i, b kj y j ) = b kj

32 32 [x, yz] = [x, y]z + y[x, z] [ϕ(x), c ϕ (β)] = i [ϕ(x), ϕ(x i )ϕ(y i )] = i [ϕ(x), ϕ(x i )]ϕ(y i ) + i ϕ(x i )[ϕ(x), ϕ(y i )] = i,j a ij ϕ(x j )ϕ(y i ) + i,j b ij ϕ(x i )ϕ(y j ) = i,j a ij ϕ(x j )ϕ(y i ) i,j a ji ϕ(x i )ϕ(y j ) = 0 Tr(c ϕ ) = n Tr(ϕ(x i )ϕ(y i )) = n β(x i, y i ) = dim g ϕ c ϕ Tr(c ϕ ) = dim g (dim g/ dim V ) g = sl 2 (K), V = K 2, ϕ g gl(v ) {x, h, y} 3.10 sl 2 -triple β(x, y) = Tr(x, y) β {x, h, y} {y, h/2, x} ( ) 3/2 0 c ϕ = xy + h 2 /2 + yx = = (dim g/ dim V ) id 0 3/2 V ϕ : g gl(v ) ϕ(g) sl(v ) g 1. ϕ(g) = ϕ(dg) = D(ϕ(g)) D(gl(V )) = sl(v ) 6.14 (Weyl ). ϕ : g gl(v ). 6.5 g W W V = W W dim W = 0 dim W > 0 dim g ϕ V 1 g W ϕ : g gl(v ) 1 g W 1 W V = W W dim W dim V/W = 1

33 2015 X V g V/W dim V/W = 1 V/W = K 0 W V K 0 W W g W 0 W/W V/W K 0 dim W/W < dim W V/W 1 g W /W (3) V/W = W/W W /W 0 W W K 0 dim W < dim W W 1 g X W = W X (4) (3), (4) W X = 0 dim V = dim W + dim X V = W X W c = c ϕ ϕ c : V V g c(w ) W Ker c g g V/W c c c W = (dim g/ dim W ) id W 0 Ker c 1 g V = W Ker c V 1 g W W V gg Hom(V, W ) V : = {f Hom(V, W ) f W } W : = {f Hom(V, W ) f W = 0} g ϕ(g)(v) W f V f W = a id W x g, w W (5) (x.f)(w) = x.f(w) f(x.w) = a(x.w) a(x.w) = 0 x.f W = 0 dim V/W = 1 1 g W V = W W W f : V W f W = id W (5) f g Ker f V g f V = W Ker f g 1 ( ) 0 t ϕ : g gl 2 (K); t 0 0

34 g gl(v ) x g gl(v ) x s x n g x g gl(v ) x s x n ad x(g) g 4.27 (iii) ad x s (g) g ad x n (g) g ad = ad gl(v ) g N := N gl(v ) (g) := {y gl(v ) ad y(g) g} x s, x n N V g W gl(v ) g W := {y gl(v ) y(w ) W, Tr(y W ) = 0} g = Dg g g W y g W 4.27 (ii), (iii) y s, y n g W g := N ( W V g W ) V g W g N g x g g g = g g g g g g h g = g h g N [g, g ] g g h y h [g, y] = 0 y W y g W Tr(y W ) = 0 y W = 0 V g y = 0 h = g ϕ : g gl(v ) x g x = s + n ϕ(x) = ϕ(s) + ϕ(n) gl(v ). ϕ(g) gl(v ) 6.16 ϕ(x) = ϕ(s) + ϕ(n) gl(v ) ad ϕ(s) ad ϕ(n) ad s ad n ad s g ad s v 1,, v m v i λ i (1 i n) ad ϕ(s)(ϕ(v i )) = ϕ(ad s(v i )) = ϕ(λ i v i ) = λ i ϕ(v i ) ad ϕ(s) ad n k (ad n) k = 0 gl(g) ad ϕ(n) gl(ϕ(g)) z g

35 2015 X V 35 (ad ϕ(n)) k (ϕ(z)) = ϕ((ad n) k z) = ϕ(0) = 0 ad ϕ(n) g 1 ( ) 0 t ϕ : g gl 2 (K); t 0 0 t g \ {0} ϕ(t) 7. sl 2 (K) g = sl 2 (K) g ( ) ( ) ( ) x =, y =, h = [x, y] = h, [h, x] = 2x, [h, y] = 2y V g h 6.17 V h V = V λ, V λ := {v V h.v = λv}. λ h V λ := {v V h.v = λv} = 0 λ K V λ V λ 0 λ V h (weight) V λ (weight space) 7.1. v V λ. x.v V λ+2, y.v V λ 2, h.v V λ h.(x.v) = [h, x].v + x.h.v = 2x.v + x.(λv) = (λ + 2)x.v. y.v h = [x, y] h.v x y x V λ 2 y V λ y V λ+2 x x y h dim V < V λ 0 V λ+2 = 0 λ λ V (highest weight) V λ λ (maximal vector) h h

36 v 0 V λ (i) h.v i = (λ 2i)v i, (ii) y.v i = (i + 1)v i+1, (iii) x.v i = (λ i + 1)v i 1 (i 0). v i := 1 i! yi.v 0 (i 0), v 1 = v i V λ 2i (i) (ii) (iii) i i = 0 ix.v i = x.y.v i 1 = [x, y].v i 1 + y.x.v i 1 = h.v i 1 + y.x.v i 1 = (λ 2(i 1))v i 1 + (λ i + 2)y.v i 2 = (λ 2i + 2)v i 1 + (i 1)(λ i + 2)v i 1 = i(λ i + 1)v i 1 i x.v i = (λ i + 1)v i 1 (i 0) 7.3. g = sl 2 (K) (m + 1) g V (i) V m (ii) m V = V m 2i. i=0 (iii) V µ 0 dim V µ = 1 (iv) x, y, h 0 m m , m 1 0, m 0 m 0 m 2... m m ϕ m. v 0 V λ 7.2 v i v 1, v 2, 0 v m0 +1 v i = 0 (i m 0 + 1) 7.2 (iii) 0 = x.v m0 +1 = (λ m 0 )v m0, v m0 0

37 2015 X V 37 λ = m v 0,, v m0 K V g V V = v 0,, v m0 K v 0,, v m0 h dim V = m 0 +1 m 0 = m Z v i V m 2i (0 i m) m V = v 0,, v m K V m 2i V V = m i=0 V m 2i, dim V m 2i = 1 {v 0,, v m } x, y, h (iv) i= g = sl 2 (K) V g V g V = W i, W i :. g dim V 0 + dim V 1 V dim V 0 + dim V 1 = (v) ϕ m. V g ϕ : g gl(v ) 7.3 i m ϕ Wi ϕ m W i ϕ(h) m, m 2,, m + 2, m 1 W i ϕ(h) (iii) 1 i dim(w i ) 0 + dim(w i ) 1 = 1 g dim V 0 + dim V 1 ϕ m dim V 0 + dim V 1 = 1 ϕ m sl 2 (K) ϕ m (m 0) ϕ m ϕ 0 = 0, ϕ 1 = id, ϕ 2 = ad m = 0, 1 m = 2 sl 2 (K) { x, h, y} 3.10 ad = ϕ 2 2 V s, t ϕ 1 = id : sl 2 (K) gl(v ) h(s) = s, h(t) = t W := Sym 2 (V ) ϕ 1 sl 2 (K) gl(w ) W s 2, st, t 2 h(s s) = s h(s) + h(s) s = 2s 2 h(s t) = s h(t) + h(s) t = 0 h(t t) = t h(t) + h(t) t = 2t 2

38 38 x(s s) = s x(s) + x(s) s = 0 x(s t) = s x(t) + x(s) t = s 2 x(t t) = t x(t) + x(t) t = 2st y(s s) = s y(s) + y(s) s = 2st y(s t) = s y(t) + y(s) t = t 2 y(t t) = t y(t) + y(t) t = 0 ϕ 2 ϕ 1 sl 2 (K) gl(sym m (V )) ϕ m sl 2 (K) sl 2 (K) K 2 sl 2 (K) gl(sym m (V )) 8. g g g g g toral 8.1. g toral. h g toral h x h ad h x 0 ad g x h ad g x 4.20 ad h x = ad g x h ad h x 0 λ 1 λ 1 0 y h ad x(y) = λ 1 y ad y g g = µ g(y, µ), g(x, µ) := {y g [x, y] = µy} x = x µ (x µ g(y, µ)) µxµ = [y, x] = λ 1 y g(y, 0) µ 2 x µ = [y, µx µ ] = 0 µ 0 x µ = 0 [y, x] = S M n (K) (i) S P A S P 1 AP

39 2015 X V 39 (ii) S A, B S AB = BA h toral 8.1 h 8.2 x h, h h [h, x] = α(h)x, [h, x] = α(h )x α(h), α(h ) K [h + h, x] = (α(h) + α(h ))x, [ch, x] = cα(h )x (c K) α : h K h 1 α h α h g α := {x g [h, x] = α(h)x ( h h)} h c g (h) := {x g [x, h] = 0} g 0 Φ := Φ(g, h) := {α h \ {0} g α 0} α Φ g h (root) g α (root space) g = c g (h) α Φ g α g h ( ) (root space decomposition, Cartan decomposition) 8.3. g = sl n (K) h := { h = diag(ξ 1,, ξ n ) n ξ i = 0 } h toral h toral g = c g (h) α Φ g α gl n (K) {e i,j } h = diag(ξ 1,, ξ n ) [h, e ij ] = (ξ i ξ j )e ij α ij : h ξ i ξ j α ij h h c g (h), e ij K g αij (i j) g = h i j e ij K h = c g (h), e ij K = g αij (i j), Φ = {α ij (i j)} h = c g (h) h toral

40 (i) [g α, g β ] g α+β (ii) B g α g α = β α g β (iii) B cg (h) c g (h). h h, x g α, y g β [h, [x, y]] = [[y, h], x] + [[h, x], y] = [x, [h, y]] + [[h, x], y] = [x, α(h)y] + [β(h)x, y] = (α + β)(h)[x, y] (i) (i) x g α, y g β ad(x) ad(y)g γ g α+β+γ g ad(x) ad(y) α+β 0 ad(x) ad(y) 0 B(x, y) = Tr(ad(x) ad(y)) = 0 g β g α g α β α g β dim g α dim g α α α dim g α = dim g α (ii) g α = β α g β α = 0 c g (h) = α Φ g α c g (h) c g (h) = {0} (iii) 8.5. h g toral h = c g (h) B h h. C := c g (h) g S S n S s (i) C n, C s C x C ad x(h) = (iii) (ad x) s (h) = 0, (ad x) n (h) = (ad x) s = ad x s, (ad x) n = ad x n x s, x n C (ii) C s h x C s h + Kx g 8.2 toral h h + Kx = h x h (iii) B h h h h B(h, h) = 0 h = (iii) B(h, C) = 0 (ii) C s h B(h, h) = 0 B(h, C s ) = 0 B(h, C n ) = 0 x C n (i) x C n C [x, h] = 0 ad x ad x ad h Tr(ad h ad x) = 0 B(h, x) = 0 B(h, C n ) = 0 (iv) C x C ad x x C x = x s + x n ad x s ad x n ad x s ad x n

41 2015 X V 41 (ii) x s h ad x s = 0 x n ad x n (v) h D(C) = 0 B [h, C] = 0 B(h, D(C)) = 0 (iii) (vi) C D(C) 0 (iv) C 3.38 Z(C) D(C) 0 z Z(C) D(C) \ {0} (ii) (v) z (i) z n C \ {0} 4.27 (iii) z n Z(C) B(x n, C) = (iii) (vi) C = h C h (i), (ii) C x (vi) B(x, y) = Tr(ad x ad y) = 0 ( y C) 8.4 (iii) B h h h h ; x B(x, ) α h h t α α(h) = B(t α, h) Φ {t α α Φ} 8.6. α Φ (i) Φ K = h (ii) α Φ (iii) x g α, y g α [x, y] = B(x, y)t α (iv) [g α, g α ] = Kt α (v) α(t α ) = B(t α, t α ) 0 (vi) x α g α \ {0} y α g α {x α, y α, h α } sl 2 -triple h α := [x α, y α ] (vii) h α = 2t α /B(t α, t α ); h α = h α. (i) Φ K h h (h ) \ {0} = h \ {0} h Φ = 0 α Φ α(h) = 0 α Φ [h, g α ] = 0 [h, h] = 0 [h, g] = 0 h Z(g) g Z(g) = 0 (ii) α Φ α Φ g α = (ii) B(g α, g) = 0 B 5.13 (iii) x g α, y g α h h B(h, [x, y]) = B([h, x], y) = B(α(h)x, y) = α(h)b(x, y) = B(t α, h)b(x, y) = B(h, B(x, y)t α ) B(h, [x, y] B(x, y)t α ) = (i) [x, y] c g (h) = h [x, y] B(x, y)t α h 8.5 B h h [x, y] = B(x, y)t α

42 42 (iv) B(g α, g α ) 0 (iii) x g α \ {0} B(x, g α ) = 0 (ii) B(x, g) = 0 B B(x, y) 0 y g α (v) α(t α ) = B(t α, t α ) t α α(t α ) = 0 [t α, g α ] = [t α, g α ] = 0 (iv) B(x, y) 0 x g α, y g α B(x, y) = 1 [x, y] = t α S := x, y, t α K S 3 S = ad S gl(g) D(S) = Kt α 4.5 t α t α h ad t α ad t α = 0 t α Z(g) = 0 (vi) (iii)-(v) x α g α \ {0} y α g α B(x α, y α ) = 2/B(t α, t α ) h α := 2t α /B(t α, t α ) [x α, y α ] = h α, [h α, x α ] = 2x α, [h α, y α ] = 2y α {x α, y α, h α } sl 2 -triple (vii) h α = [x α, y α ] = B(x α, y α )t α {x α, y α, h α } sl 2 -triple [h α, x α ] = 2x α [h α, x α ] = [B(x α, y α )t α, x α ] = B(x α, y α )[t α, x α ] = B(x α, y α )α(t α )x α B(x α, y α ) = 2/B(t α, t α ) t α = t α 8.6 (vi) sl 2 -triple s α 8.7. (i) α Φ dim g α = 1 (ii) α, cα Φ (c K) c = ±1 (iii) α, β Φ β(h α ) Z β β(h α )α Φ (iv) α, β, α + β Φ [g α, g β ] = g α+β (v) α, β Φ β ±α r, q Z β rα, β + qα Φ β + iα Φ ( r i q) β(h α ) = r q (vi) g g α (α Φ). (i), (ii) α Φ sl 2 -triple s α = x α, y α, h α K sl 2 g g sl 2 sl 2 sl ϕ m g ad h α cα Φ 2c = cα(h α ) Z m/2 (m Z) c > 1 (1/c)(cα) = α Φ cα (1/c) 1/c = m/2 m Z 1 > 1/c = m/2 c = ±2 c < 1 c = ±1/2 α Φ ±α, ±α/2, ±2α

43 2015 X V 43 α/2 = (1/4)2α α/2 2α 2α s := Ky α Kh α g α g 2α s g h α Ds Tr(ad s h α ) = 0 α(h α ) = 2 Tr(ad s h α ) = α(h α ) α(h α ) dim g α + 2α(h α ) dim g 2α = dim g α + 4 dim g 2α dim g α + 2 dim g 2α = 1 dim g α 1 dim g α = 1, dim g 2α = 0 2α 2(α/2) = α Φ α/2 (i) (ii) (iii)-(v) α, β Φ α(h α ) = 2 β = ±α (iii) (iv) α + β Φ β ±α (iii)-(v) β ±α V := i Z g β+iα (i) g β+iα dim g β+iα = 1 β ±α (ii) β +iα 0 V g s α β(h α )+2i i β + iα Φ β(h α ) Z α 0 1 β(h α ) + 2i 7.4 V q, r 7.3 β + iα Φ ( r i q) (β rα)(h α ) = (β + qα)(h α ) (v) β(h α + 2q) 7.2 [g α, g β ] 0 (vi) B h h h ; x B(x, ) h h λ, µ h (λ, µ) := B(t λ, t µ ) h 8.6 (i) Φ {α 1,, α l } h 8.8. β Φ β = l j=1 c jα j c j Q. i (α i, β) = l j=1 (α i, α j )c j 2/(α i, α i ) 2(α i, β) l (α i, α i ) = j=1 2(α i, α j ) (α i, α i ) c j.

44 44 B h h {α i } h det((α i, α j )) 0 ( ) ( l ) 2(αi, β) 2 det = det((α i, α j )) 0. (α i, α i ) (α i, α i ) Φ Q h E Q E Q Q l h = E Q Q K 8.9. h (, ) E Q. λ, µ h (λ, µ) = B(t λ, t µ ) = Tr(ad t λ ad t µ ) = α Φ α(t λ )α(t µ ) = α Φ(α, λ)(α, µ). β Φ (β, β) = α Φ (α, β) 2 (β, β) 2 1 (β, β) = α Φ (α, β) 2 (β, β) (iii) 2(α,β) Z (β,β) 2 (β, β) Q α, β Φ (α, β) = (β, β) 2 2(α, β) (β, β) Q E Q Q (, ) E Q λ (λ, λ) = α Φ(α, λ) 2 0. (λ, λ) = 0 α Φ (α, λ) = 0 λ g = 0 E := E Q Q R (, ) E (, ) E (E, (, )) (E, Φ, (, )) (R1) Φ E E 0 (R2) k α, kα Φ k = ±1. (R3) α Φ σ α σ α (Φ) Φ (R4) α, β Φ β, α.

45 2015 X V E (α, β) R α E α E P α P α = {β E (β, α) = 0} σ GL(E) σ E (reflection) (i) α E σ P α (ii) σ(α) = α σ σ α σ α (β) = β 2(β, α) (α, α) α. E E = Rα P α β = cα + γ (c R, γ P α ) (β, α) β α c = (α, α) σ α (β) = σ α (cα + γ) = cσ α (α) + σ α (γ) = cα + γ = β 2cα 2(β, α) (α, α) β, α c α,β 9.3. Φ E α Φ σ α (Φ) = Φ σ GL(E) (i) σ(φ) = Φ. (ii) σ(α) = α. (iii) P σ σ = σ α P = P α. τ := σ σ α τ(α) = α, τ(φ) = Φ τ Rα τ E/Rα E = Rα P β E β = cα + γ (c R, γ P ) τ(β) = σ(β β, α α) = σ(β) β, α σ(α) = γ + ( β, α c)α τ E/Rα τ 1 τ f τ (x) (x 1) l (l = dim E) τ(φ) = Φ β, τ(β),, τ k (β) (k Φ ) k 0 τ k 0 (β) = β k

46 46 β τ k (β) = β Φ E τ k = 1 τ f τ (x) x k 1 f τ (x) = x 1 τ = E Φ Φ : (R1) Φ E E 0 (R2) k α, kα Φ k = ±1. (R3) α Φ σ α σ α (Φ) Φ (R4) α, β Φ β, α. Φ E Φ σ α (α Φ) Φ W Φ W. Φ = {α 1,, α d } w W w(φ) = Φ W Sym(Φ) W Sym(Φ); w [i j if w(α i ) = α j ] Φ E Sym(Φ) 9.6. Φ E W σ GL(E) σ(φ) = Φ α, β Φ (i) σσ α σ 1 = σ σ(α). (ii) β, α = σ(β), σ(α) β Φ σσ α σ 1 (σ(β)) = σ(σ α (β)) = σ(β β, α α) = σ(β) β, α σ(α) σ σ(α) (σ(β)) = σ(β) σ(β), σ(α) σ(α) 9.7. (E, Φ), (E, Φ ) ϕ : E E ϕ(φ) = Φ ϕ(β), ϕ(α) = β, α ( α, β Φ) (E, Φ) (E, Φ ) Φ Φ (isomorphic) (E, (, ) E, Φ), (E, (, ) E, Φ ) ϕ : E E ϕ(φ) = Φ (E, (, ) E, Φ), (E, (, ) E, Φ ) (E, (, ) E, Φ) c, d R \ {0} (E, c(, ) E, dφ). (E, (, ) E, Φ)c, d R\{0} (E, c(, ) E, dφ) α, β Φ β, α E =

47 2015 X V 47 ϕ(β), ϕ(α) E E E ϕ GL(E) 9.6 ϕ : E E ϕ(φ) = Φ ϕ(β), ϕ(α) = β, α ϕ Φ Φ W, W W W ; σ ϕ σ ϕ 1 Φ Aut(Φ) := {g GL(E) g(φ) = Φ} 9.6(i) W Aut(Φ) (E, Φ) E E R l E R l α, β Φ θ α ±β β α ( α, β, β, α, θ, β 2 / α 2 ) α, β β, α θ β 2 / α π/2 1 1 π/ π/ π/ π/ π/ π/6 3. (α, β) = α β cos θ β, α = 2(β, α) (α, α) = 2 β α cos θ α, β β, α = 4 cos 2 θ 0 cos 2 1 α, β, β, α 0 α, β β, α 4 α ±β α, β β, α = 4 θ = π/2 α, β β, α = 0 α, β β, α = 1, 2 3 β α β, α α, β = β 2 α 2 1

48 48 α, β β, α = 2 ( α, β β, α ) = (2, 1) ( 2, 1) cos 2 θ = 1/2 θ = π/4 3π/4 β 2 / α 2 = α, β Φ (α, β) > 0 α β (α, β) < 0 α + β. β β (α, β) > α, β β, α 1 α, β = 1 σ α (β) = α β Φ β, α = 1 β α Φ σ β α (β α) = α β Φ A 1 A 1, A 2, B 2, G 2. dim E = 2 Φ θ α 1, α 2 Φ α 1, α 2 Φ θ π/2 9.9 θ π/2, π/3, π/4, π/6α 3 := σ α2 ( α 1 ) α 2 α 1 α 4 := σ α3 ( α 2 ) α 3 α 2 2π/θ θ Φ θ π/2, π/3, π/4, π/6 A 1 A 1, A 2, B 2, G α, β Φ p, q 0 i Z β + iα Φ q i p. I := {i Z β + iα Φ} I p, q β I p, q 0 q i p β + iα Φ 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 (α, β + (r + 1)α) 0, (α, β + (s 1)α) 0 (r s + 2)(α, α) {β + iα Φ i Z} = {β + iα q i p} β α (α-string through β) α, β Φ E E E Φ E

49 X V Φ E Φ (B1) E. (B2) β Φ β = Σk α α (α ) k α α k α 0 k α 0 (B2) β htβ := Σk α β (height) α k α 0 β α k α 0 β Φ + Φ ξ, η E ξ η = Σk α α α k α 0 ξ η E Φ α β (α, β) 0. (α, β) > 0 α ±β 9.10 α β Φ (B2) γ E Φ + (γ) := {α Φ (γ, α) > 0} γ E \ α Φ P α γ (regular) (singular) γ Φ = Φ + (γ) Φ + (γ) α Φ + (γ) α = β 1 + β 2 β 1, β 2 Φ + (γ) α (decomposable) (indecomposable) γ E (γ) Φ + (γ) (γ) Φ. 5 1 Φ + (γ) (γ) Z (γ) Z Φ + (γ) γ α α (γ) α α = β 1 + β 2 β 1, β 2 Φ + (γ) (γ, α) = (γ, β 1 ) + (γ, β 2 ) (γ, β 1 ), (γ, β 2 ) > 0 (γ, α) β 1, β 2 (γ) Z 2 α β (γ) (α, β) 0 (α, β) > 0 β ±α 9.10 α β Φ α β Φ + (γ) β α Φ + (γ) α = β + (α β) α β 3 (γ) r α α = 0 (α (γ), r α R) r α s α α = t β β s α, t β 0

50 50 α β ϵ = s α α (ϵ, ϵ) = α,β s αt β (α, β) 0 2 ϵ = 0 0 = (γ, ϵ) = s α (γ, α) s α = 0 t β = 0 E 1 S S 2 S 4 (γ) Φ 1 Φ = Φ + (γ) Φ + (γ) (B2) (γ) E 3 (γ) (B1) 5 Φ γ E = (γ) α (γ, α) > 0 γ E (B2) γ Φ + Φ + (γ) Φ Φ + (γ) Φ = Φ + Φ, Φ = Φ + (γ) Φ + (γ) Φ + = Φ + (γ) Φ = Φ + (γ) Φ + = Φ + (γ) (γ) = (γ) = dim E = (γ) E \ α Φ P α (Weyl chamber) γ E C(γ) C(γ) = C(γ ) P α (α Φ) γ, γ Φ + (γ) = Φ + (γ ) (γ) = (γ ) W { } { }; (σ, C(γ)) C(σ(γ)), W { } { }; (σ, ) σ( ) W α Φ + (γ) (σ(α), σ(γ)) = (α, γ) σ(φ + (γ)) = Φ + (σ(γ)) σ( (γ)) = (σ(γ)) { } { }; (γ) C(γ). = (γ) C( ) := C(γ) (fundamental Weyl chamber relative to ) Φ α Φ + \ β α β Φ +. α = γ s γγ α s γ 0 γ (α, γ) 0 (α, α) = γ s γ (α, γ) 0 α = 0 β (α, β) > α β Φ α β

51 2015 X V 51 γ β s γ > 0 α β γ (B2) α β β Φ + α 1,, α k (i < j α i = α j ) β = α α k α α i Φ ( i) htβ α σ α (Φ + \ {α}) = Φ + \ {α}. β = γ s γγ Φ + \ {α} (s γ 0) β ±α γ α s γ > 0 σ α (β) = β β, α α γ s γ (B2) σ α (β) σ α (α) = α δ := 1 2 β Φ β α σ + α (δ) = δ α α 1,, α t i < j α i = α j σ i := σ αi σ 1 σ t 1 (α t ) Φ s (1 s < t) σ 1 σ t = σ 1 σ s 1 σ s+1 σ t 1. β i = σ i+1 σ t 1 (α t ) (0 i t 2), β t 1 = α t β 0 Φ, β t 1 Φ + β s Φ + s σ s (β s ) = β s 1 Φ 10.7 β s = α s 9.6 σ W σ σ(α) = σσ α σ 1 σ s = σ αs = σ βs = σ σs+1 σ t 1 (α t ) = = (σ s+1 σ t 1 )σ t (σ s+1 σ t 1 ) 1. σ 1 σ t = (σ 1 σ s 1 )σ s (σ s+1 σ t ) = (σ 1 σ s 1 )(σ s+1 σ t 1 ) σ W σ = σ 1 σ t σ σ(α t ) Φ. σ(α t ) = σ 1 σ t (α t ) = (σ 1 σ t 1 )(α t ) σ(α t ) Φ + (σ 1 σ t 1 )(α t ) Φ Φ W

52 52 (i) γ E σ W α (σ(γ), α) > 0 (ii) Φ σ( ) = σ W (iii) α Φ σ(α) σ W (iv) W = σ α α. (v) σ W σ( ) = σ = 1 (i), (ii). W := σ α α W (i) (iii) W W = W (i) δ = 1 2 α Φ α (σ(γ), δ) σ W α + σ α σ W σ 10.8 (σ(γ), δ) (σ α σ(γ), δ) = (σ(γ), σ α (δ)) = (σ(γ), δ α) = (σ(γ), δ) (σ(γ), α) α (σ(γ), α) 0 γ (σ(γ), α) 0 (σ(γ), α) = 0 γ P σ 1 α (σ(γ), α) > 0 (ii) 10.4 (i) (iii) (ii) α γ P α \ β Φ\{±α} P β γ γ (γ, α) = ϵ > 0 (γ, β) > ϵ ( β ±α) α Φ + (γ ) α (iv) W = W α Φ σ α W (iii) β = σ(α) σ W σ β = σ σ(α) = σσ α σ 1 σ(α) = σ 1 σ β σ W (v) σ( ) = σ 1 (iv) σ σ( ) = σ W σ = σ α1 σ αl. l l(σ) σ (length) l = l(σ) σ (reduced expression) n(σ) := {α Φ + σ(α) Φ } σ W l(σ) = n(σ). l(σ) l(σ) = 0 σ = 1 n(σ) = 0 l(τ) < l(σ) τ W l(τ) = n(τ)

53 2015 X V 53 σ = σ α1 σ αl α := α l σ(α t ) Φ 10.7 n(σσ α ) = n(σ) 1 l(σσ α ) = l(σ) 1 < l(σ) l(σσ α ) = n(σσ α ) l(σ) = n(σ) λ, µ C( ) σ W σ(λ) = µ λ = µ. l(σ) l(σ) = 0 σ = 1 l(σ) > σ α σ(α) Φ 0 (µ, σα) = (σ 1 µ, α) = (λ, α) 0 (λ, α) = 0 σ α λ = λ (σσ α )(λ) = µ l(σσ α ) = l(σ) 1 λ = µ σ = σ α1 σ αl α := α l σ(α t ) Φ 10.7 n(σσ α ) = n(σ) 1 l(σσ α ) = l(σ) 1 < l(σ) l(σσ α ) = n(σσ α ) l(σ) = n(σ) Φ Φ 1, Φ 2 Φ Φ = Φ 1 Φ 2 (Φ 1, Φ 2 ) = 0 Φ (reducible) (irreducible) 2 A 1 A 1, A 2, B 2, G 2 4 A 1 A Φ Φ 1, 2 = 1 2 ( 1, 2 ) = 0. Φ Φ 1, Φ 2 Φ Φ = Φ 1 Φ 2 (Φ 1, Φ 2 ) = 0 Φ 1 (, Φ 2 ) = 0 E (E, Φ 2 ) = 0 Φ 2 = Φ 1 Φ 2 = ( Φ 1 ) ( Φ 2 ) Φ = 1 2 ( 1, 2 ) = 0 Φ i := W ( i ) (i = 1, 2) (iii) Φ = Φ 1 Φ 2 α i i (i = 1, 2) (σ α1 σ α2 σ α2 σ α1 )(γ) = 4(α 1, α 2 ) α 1 2 α 2 2 ((γ, α 2)α 1 (γ, α 1 )α 2 ) = 0 α i i (i = 1, 2) σ α1 σ α2 = σ α2 σ α1 (α 1, α 2 ) = 0 σ α2 (α 1 ) = α 1 Φ i = W ( i ) = σ α α i ( i ) σ α α i ( i ) i Φ i i (Φ 1, Φ 2 ) = 0 Φ Φ i0 = (i 0 {1, 2}) i0 =

54 Φ β = k α α α β htα < htβ k α > 0. S S x x x < x 1 x 1 S x 1 x 1 < x 2 x 2 S x < x 1 < x 2 < S (Φ, ) β = k α α (α ) β 0 α α β β 0 1 = {α k α > 0}, 2 = {α k α = 0} = α 2 (α, β) 0 Φ α 2 ( 1, α) 0 α 1 (α, α ) < 0 (α, β) < β + α Φ β 2 = α k α > 0 β β α (α, β) 0 = E α (α, β) > 0 (β, β) > β β Φ β β β β Φ W E W {0} E α W E. W W (α) W (W (α)) = W (α) W E E 0 W E E E = E E x E, y E, w W (wx, y) = (x, w 1 y) = 0 wx (E ) = E E W E W α Φ σ α (E ) = E α E E P α α E E Φ Φ E 0 E = E Φ Φ 2 W. α, β, γ Φ W (α) E σ W (σ(α), β) 0 α σ(α) (α, β) 0 σ W (σ(γ), β) 0 (γ, β) 0

55 2015 X V β 2 α 2, β 2 γ 2 {2, 3, 1 2, 1 3 } γ 2 α = β 2 γ 2 2 α 2 β {2, 3, 1 2 2, 1 3 } α, β Φ σ W σ(α) = β α = ±β α ±β (α, β) α, β = β, α = ±1 β β = σ β (β) α, β = 1 (σ α σ β σ α )(β) = σ α σ β (β α) = σ α ( β α + β) = α Φ 2 (long root) (short root) (long root) Φ β. α Φ (β, β) (α, α) W α C( ) β α 0 γ C( ) (γ, β α) 0 γ = β γ = α (β, β) (β, α) (α, α) Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and Engineering, Saitama University. Shimo-Okubo 255, Sakura-ku Saitama-shi, , Japan. address: kwatanab@rimath.saitama-u.ac.jp