OCAMI 2015 12 16
1. 2. R 3. Lie 4. U(n), SU(n), O(n), SO(n), Sp(n) 5. U(n), SU(n), O(n), SO(n), Sp(n) 6. Lie 7. G 2 G 2 /SO(4)
1. M Riemann M Riemann def x M, s x : s.t. s 2 x = id, x s x s x x R n S n T n S M antipodal set def x, y S s x (y) = y S great antipodal set def S = max{ A A M } =: # 2 M M 2-number
1 M = R n {x} x R n 2 M = S n {x, x} x S n R n+1 3 M = RP n {Re 1,..., Re n+1 } e 1,..., e n+1 R n+1 o.n.b. 4 M = T n # 2 T n = 2 n I(M) M I(M) M M G := I 0 (M) M = G/K K := {g G go = o} (o M)
Ko = o K T o M R Riemann G/K Riemann Hermite R G/K Lie Lie SO(n) U(n) Sp(n) Grassmann G k (K n ) K = R, C, H) S n R Hermite T.- 2012
2. R 1 T.- 2013 R M 1 M 2 M I 0 (M) 3 M Weyl W R Hermite ˆM = Ĝ/ ˆK M = F (τ, ˆM) τ ˆM W Ĝ I τ : g τgτ 1 Weyl
R Riemann Grassmann G k (R n ) 2k n G 1 (R n ) = S n 1 G 2 (R n ) = Q n 2 (C) 2 R k 3 R 2013 k = 3, 4 G k (R n ) Lie R
3. Lie Lie G Riemann Riemann x G s x (y) = xy 1 x (y G) 1 G s 1 (y) = y y 2 = 1 x 2 = 1, y 2 = 1 s x (y) = y xy = yx 1 S G S = Z 2 Z 2 r S = 2 r r rank(g) r > rank(g)
4. U(n), SU(n), O(n), SO(n), Sp(n) n := ±1... ±1 O(n) ± n := {g n det g = ±1} O(n), U(n), Sp(n) n SO(n), SU(n) + n
5. U(n), SU(n), O(n), SO(n), Sp(n) U(n) Z = id {z C z = 1} Z µ Z µ U(n)/Z µ U(n) Lie SU(n) = id {z C z n = 1} = Z n Z µ Z n µ µ n SU(n)/Z µ SU(n) Lie D[4] := ±1 0 0 ±1, 0 ±1 ±1 0 O(2) D[4]
D ± [4] := {g D[4] det g = ±1} n n = 2 k l, l 0 s k C(s, n) := D[4] D[4] n/2 s O(n) s D[4] n/2 s 2 T.- µ Z µ U(n) µ θ 1 2µ π n : U(n) U(n)/Z µ U(n)/Z µ
1 n µ π n ({1, θ}c(0, n)) = π n ({1, θ} n ) 2 n µ π n ({1, θ}c(s, n)) (0 s k) (s, n) = (k 1, 2 k ) 2 D[4] C(k 1, 2 k ) = D[4] D[4] 2 D[4] D[4] D[4] = C(k, 2 k ) C(k 1, 2 k )
3 T.- µ n Z µ SU(n) µ θ 1 2µ π n : SU(n) SU(n)/Z µ SU(n)/Z µ 1 n µ π n ( + n ) 2 n µ a k = 1 π n ( + n θ n ), π n ((D + [4] θd [4]) l )
n = µ = 2 π 2 ( + 2 θ 2 ) b k 2 µ = 2 k l, l b1 k = k π n ( + n θ n ), π n(c(s, n)) (1 s k) (s, n) = (k 1, 2 k ) b2 1 k < k π n ({1, θ} + n ), π n({1, θ}c(s, n)) (1 s k) (s, n) = (k 1, 2 k ) n = 4 π 4 ({1, θ} + 4 ) + 4 = 2 2 D[4] D[4] = C(2, 4) π 4 ({1, θ} + 4 )
O(n) = {±1 n } = Z 2 O(n)/{±1 n } O(n) Lie n 2 n SO(n) SO(n) n {±1 n } n {1 n } n SO(n)/{±1 n } SO(n) Lie Sp(n) = {±1} = Z 2 Sp(n)/{±1 n } Sp(n) Lie
±1 : Q[8] := {±1, ±i, ±j, ±k} 4 T.- n 2 2 k l 2 k l I O(n)/{±1 n } π n (C(s, n)) (0 s k) (s, n) = (k 1, 2 k ) II n SO(n)/{±1 n }
1 k = 1 π n ( + n ), π n (D + [4] l ) n = 2 π 2 ( + 2 ) 2 k 2 π n ( + n ), π n (C(s, n)) (1 s k) (s, n) = (k 1, 2 k ) n = 4 π 4 ( + 4 ) III Sp(n)/{±1 n } π n (Q[8] C(s, n)) (0 s k) (s, n) = (k 1, 2 k )
6. Lie G Lie Z G G G/Z = Inn(g) G Lie g Inn(g) g 3 4 II III g = su(n), so(n), sp(n) Inn(g) Aut(g) g Inn(g) G G/Z Ad
5 T.- n 2 2 k l 2 k l I τ : su(n) su(n) ; X X Aut(su(n)) {e, τ}ad(c(s, n)) (0 s k) (s, n) = (k 1, 2 k ) II Aut(so(n)) Ad(C(s, n)) (0 s k) (s, n) = (k 1, 2 k )
III Aut(sp(n)) Ad(Q[8] C(s, n)) (0 s k) (s, n) = (k 1, 2 k )
7. G 2 G 2 /SO(4) Lie G 2 Cayley G 2 G 2 = {e} F (s e, G 2 ) = {g G 2 g 2 = e} = {e} M 1 + M 1 + = G 2 /SO(4) o M 1 + F (s o, M 1 + ) = {o} M 1,1 + M 1,1 + = (S 2 S 2 )/Z 2 x Z 2 = {±1} x (p, q) = (xp, xq) S 2 S 2 π : S 2 S 2 (S 2 S 2 )/Z 2
[p, q] := π(p, q) ( (p, q) S 2 S 2) e 1, e 2, e 3 S 2 S 2 S 2 f 1, f 2, f 3 S 2 S 2 S 2 (S 2 S 2 )/Z 2 A := {[e 1, ±f 1 ], [e 2, ±f 2 ], [e 3, ±f 3 ]} A M 1,1 + A 1,1
6 T.- 1 G 2 /SO(4) {o} A 1,1 7 2 G 2 {e, o} A 1,1 8 3 Aut(g 2 ) Aut(g 2 ) = Inn(g 2 ) = G 2 2