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1 R-space ( ) Version 1.1 (2012/02/29)

2 i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

3 ii 1 Lie Killing Cartan Lie Lie Lie Lie s s R-space Lie R-space s- R-space R-space Kähler C s Hermite Kähler C-space

4 iii 6.3 Hermite

5 1 1 Lie, Lie,. 1.1 Killing, g Lie Killing 1.1 B : g g R Killing : B(X, Y ) := tr(ad X ad Y ). (1.1) 1.2 Killing B, : B([X, Y ], Z) + B(Y, [X, Z]) = 0 ( X, Y, Z g). (1.2) Killing 1.3 Lie gl n (R) Killing B : B(X, Y ) = 2ntr(XY ) 2tr(X)tr(Y ). (1.3) 1.4 Lie g g, Killing B, B. B B,, B = B g g. (1.4) 1.5 Lie sl n (R) Killing B : B(X, Y ) = 2ntr(XY ). (1.5)

6 2 1 Lie 1.6 Lie so(p, q) Killing Killing 1.7 ([6], p50, Theorem 1.45) Lie, Killing. 1.8 ([6], pp , Corollary 4.26, 4.27) Lie, Killing. 1.2 Cartan, g Lie, B g Killing Cartan θ : g g *1, θ Lie, θ 2 = id. 1.9 θ : g g Cartan, B θ : B θ (X, Y ) := B(X, θ(y )) (for X, Y g). (1.6) Cartan θ, ± Cartan θ g = k p Cartan., k 1, p Cartan g = k p : [k, k] k, [k, p] p, [p, p] k. (1.7) Cartan 1.12 Lie sl n (R), Cartan : θ(x) := t X. (1.8) *1 involution

7 1.3 3 Cartan sl n (R) = k p, : k = so(n), p = {X sl n (R) X = t X}. (1.9) 1.13 Lie so(p, q), Cartan : θ(x) := I p,q XI p,q. (1.10) Cartan so(p, q) = k p, : {( ) } k = so(p) so(q), p = t C C M C q,p (R). (1.11) Cartan 1.14 Lie g, id Cartan., Cartan, ([6], p358, Corollary 6.18, 6.19) Lie g, Cartan., Cartan ( ). 1.3, g Lie, B Killing, θ Cartan, g = k p Cartan X p, ad X B θ. ad X.,., a p., a a α a, α : g α := {X g [H, X] = α(h)x ( H a)}. (1.12)

8 4 1 Lie 1.18 α a (a ), : α 0, g α 0.,, = (g, a) = (g, a), : (1) g = g 0 ( α g α) B θ, (2) [g α, g β ] g α+β ( α, β {0}), (3) θ(g α ) = g α ( α {0}). (1) E ij sl n (R), Cartan, : (1) a := { a k E kk tr = 0} p, (2) g 0 = a, (3) g εi ε j = span{e ij }, ε i ( a k E kk ) := a i, (4) = {ε i ε j i j} sl n (C), Cartan, : (1) a := { a k E kk tr = 0} p, (2) g 0 = a, (3) g εi ε j = span{e ij }, ε i ( a k E kk ) := a i, (4) = {ε i ε j i j} so(1, n), Cartan, : (1) a := span{e 12 + E 21 } p, (2) g 0 = so(n 1) a, (3) α a α(e 12 + E 21 ) = 1, = {±α} so(2, 2), Cartan, : (1) a := span{e 13 + E 31, E 24 + E 42 } p, (2) g 0 = a, (3) = {±ε 1 ± ε 2 }, ε 1, ε 2 a : ε i (a 1 (E 12 + E 21 ) + a 2 (E 24 + E 42 )) = a i. (1.13)

9 so(2, n) ( n > 2), Cartan, : (1) a := span{e 13 + E 31, E 24 + E 42 } p, (2) g 0 = so(n 2) a, (3) = {±ε 1 ± ε 2, ±ε 1, ±ε 2 }, ε 1, ε 2 a ([6], p378, Theorem 6.51) a, a p., K. K k Lie. a, g., G/K. Cartan, ([6], p380, Corollary 6.53) = (g, a), Λ := {α 1,..., α r }, : (i) Λ a, (ii) α, c 1,..., c r Z 0 c 1,..., c r Z 0 : α = c 1 α c r α r. 1.28, Λ = {α 1,..., α r }, α = c 1 α 1 + +c r α r. (1) α, c 1,..., c r Z 0. (2) α, c 1,..., c r Z 0. (3) α, : c 1α c rα r, c 1 c 1,..., c r c r.

10 6 1 Lie sl n (R), α i := ε i ε i+1 (i = 1,..., n 1), {α 1,..., α n 1 }., α = ε 1 ε n = α α n so(2, 2), α 1 = ε 1 ε 2, α 2 = ε 1 + ε 2, {α 1, α 2 } so(2, n) ( n > 2), α 1 = ε 1 ε 2, α 2 = ε 2, {α 1, α 2 }., α = ε 1 + ε 2 = α 1 + 2α = (g, a), (, ).,, Weyl ([6], p383, Theorem 6.57) N K (a) Z K (a), K a., N K (a)/z K (a), Weyl.

11 7 2 Lie Lie, Lie Lie,.,. Lie, graded Lie algebra, Lie. g Lie. 2.1 Lie Lie 2.1 g = k Z gk Lie, : [g p, g q ] g p+q ( p, q Z). 2.2 Lie g = k Z gk ν, : (i) g ν 0, (ii) p > ν g p = Lie 2.3 sl 3 (R) sl 3 (C), 1 Lie : g 1 :=, g0 :=, g1 := (2.1)

12 8 2 Lie Lie 2.4 sl 3 (R) sl 3 (C), 1 Lie : g 1 :=, g0 :=, g1 := (2.2) 2.5 sl 3 (R) sl 3 (C), 2 Lie : g 2 :=, g 1 :=, g0 :=, g 1 :=, g2 :=. (2.3) 2.6, sl n (R) sl n (C), Lie Lie 2.7 Lie g = k Z gk, Z g, : g k = {X g [Z, X] = kx} ( k Z). (2.4) Lie, Z g., = (g, a), Λ = {α 1,..., α r }. 2.8 a {H 1,..., H r } Λ, : α i (H j ) = δ ij ( i, j). 2.9 Z := n 1 H n r H r ( n 1,..., n r Z 0 )., : (1) ad Z Lie, Z. (2) α, (1) Lie α(z). Z Lie, : g k = g α. (2.5) α(z)=k

13 2.1 Lie Lie 2.10 sl 3 (R), Λ = {α 1, α 2 }., (1) Λ : H 1 := 1 2 1, H 2 := , (2.6) (2) Z = H 1, H 2 1 Lie, (3) Z = H 1 + H 2 2 Lie so(1, n), Z = E 12 + E 21 1 Lie. : g 1 = g α, g 0 = g 0, g 1 = g α. (2.7) 2.12 Lie, so(n + 1, C) 1 Lie g C = (g 1 ) C (g 0 ) C (g 1 ) C so(2, n) ( n > 2), (1) Z = H 1 1 Lie, (2) Z = H 2 2 Lie, (3) Z = H 1 + H 2 3 Lie Lie Lie., Lie, 2.9, Lie g = k Z gk,., Lie Der(g). Z, Z p ( Cartan g = k p ) (cf. [8]) Lie g = k Z gk Z., Cartan θ : θ(z) = Z. θ(z) = Z, : θ(g k ) = g k ( k). gradereversing., a Λ, ( ).

14 10 2 Lie Lie 2.2 Lie Lie 2.16 Lie q g, : Lie g = k Z gk, q = k 0 gk. Lie, Lie Lie g = k Z gk Z a., Lie q : q = g 0 g α. (2.8) α(z) Lie Lie, Lie,. [6] Lie.,.

15 s s- g = k p Cartan, Lie (G, K). 3.1 K p s- : ϕ : K GL(p) : g Ad g p. (3.1) s- 3.2 sl n (R) = so(n) Sym 0 n(r) s-, SO(n) Sym 0 n(r) ; g.x := gxg 1. (3.2) 3.3 sl n (C) = su(n) Herm 0 n(c) s-, SU(n) Herm 0 n(c) ; g.x := gxg 1. (3.3) s- G/K, g K, g : G/K G/K : [h] [gh] (3.4) o = [e] (g ) o : T o (G/K) T o (G/K) (3.5).

16 , s-,., s-,, s-,. 3.2 s- s-, v p K.v α, : k g α = 0, p g α = α, k p : k α := k (g α g α ), p α := p (g α g α ). (3.6) 3.7 sl 3 (R), α = ε 1 ε 2, : 1 k α = k = span 1 1 p α = p = span 1,. (3.7) 3.8 : (1) k 0 := k g 0 Lie, (2) α, k α = k α, p α = p α. 3.9 α, : k α p α = g α g α. 3.10, : (1) dim k α = dim p α = dim g α ( α ), (2) ( k p ): k = k 0 ( α>0 k α), p = a ( α>0 p α). (3.8)

17 3.2 s s-, K v Lie k v v p, : k v = {X k [X, v] = 0}. (3.9) 3.12 Z a, : k Z = k 0 k α. (3.10) α(z)= sl 3 (R) = so(3) Sym 0 3(R) s-, : (1) Z = H 1, so(3) Z = so(2) (= so(1) so(2)), (2) Z = H 2, so(3) Z = so(2) (= so(2) so(1)), (3) Z = H 1 + H 2, so(3) Z = so(1, n) = so(n) p s-, k = so(n)., 0 Z a, : k Z = k 0 = so(n 1) so(2, n) = so(2) so(n) p s-, k = so(2) so(n)., : (1) Z = H 1, k Z = k 0 k α2 = so(n 1), (2) Z = H 2, k Z = k 0 k α1 = so(n 2) so(2), (3) Z = H 1 + H 2, k Z = k 0 = so(n 2) s s-, : {c 1 H c r H r a c 1,..., c r 0}. (3.11) 3.17 s-, k Z, 2 r.

18 14 4 R-space 4.1 Lie R-space R-space 4.1 q g Lie., G g Lie Lie, Q := N G (q)., G/Q R-space R-space (1) 1 k 1 < < k s < n, (k 1,..., k s ) F k1,...,k s (R n ). F k1,...,k s (C n ). 4.2 sl 3 (R) 3 Lie R-space, : G 1 (R 3 ), G 2 (R 3 ), F 1,2 (R 3 ). 4.3 sl 3 (C) 3 Lie R-space, : G 1 (C 3 ), G 2 (C 3 ), F 1,2 (C 3 ). 4.4 F k1,k 2,...,k s (R n ) R-space.

19 4.2 s- R-space s- R-space s- R-space 4.5 R-space, s-.., s- R-spece R-space (2) 4.6 sl 3 (R) 3 R-space, : SO(3) Sym 0 3(R) := {X sl 3 (R) t X = X}, g.x := gxg 1. (4.1) 4.7 sl 3 (C) 3 R-space, : SU(3) Herm 0 3(C) := {X sl 3 (C) t X = X}, g.x := gxg 1. (4.2) 4.8 so(1, n) R-space, S n so(1, n) = g 1 g 0 g 1 so(n + 1, C) Lie. R-space, SO(n + 1)/(SO(2) SO(n 1)) = G 2 (R n+1 ) (= Q n 1 (C)). 4.3 R-space Lie g = g 1 g 0 g 1 R-space ( R-space ). 1 Lie, sl 3 (R) 2 1 Lie R-space, G 1 (R 3 ) G 2 (R 3 ). 4.4 Kähler C Lie g C = k Z (gc ) k R-space Kähler ( Kähler C-space ).

20 16 4 R-space 4.13 sl 3 (C) 2 1 Lie R-space, G 1 (C 3 ) G 2 (C 3 ). Hermite. F 1,2 (C 3 ) Kähler : (1) M Kähler C-space, (2) M Lie, (3) M = G/C(S), G Lie, S G, C(S) S. (4) M Kähler.

21 17 5 R-space Lie H M.,, * H M, SO(n) R n SO(3) Sym 0 3(R) s SU(3) Herm 0 3(C) s s- 5.6 Z a *2, : α, α(z) 0. *1 cohomogeneity *2 regular element

22 Z := H H r. 5.8 Z a., s-, : Z K.Z, Z. 5.9 Cartan g = k p s-, dim a (, g ) K p s-, B θ p p SO(n) Sym 0 n(r) s-, B θ p p (X, Y ) = 2ntr(X t Y ) Z p B θ (Z, Z) = 1., : K.Z {X p B θ (X, X) = 1} = S n 1. (5.1) dim a = 2, Z a B θ (Z, Z) = 1., K.Z p S n 1.,, g = so(1, p) so(1, q) : S p 1 S q 1 S p+q 1 ( p). (5.2) 5.15 sl 3 (R) : F 1,2 (R 3 ) S 4 Sym 0 3(R) = R 5. (5.3) 5.16 sl 3 (C) : F 1,2 (C 3 ) S 7 Herm 0 3(C) = R 8. (5.4), 2 Lie., A 1 + A 1, A 2, B 2 (= C 2 ), BC 2, G 2.

23 19 6 Hermite M Hermite. N M M (real form), : M σ, N = Fix(σ; M) R n C n. 6.3 RP n CP n. 6.2 Kähler C-space 6.4 Lie g = k Z gk R-space G/Q, g C = k Z (gk ) C Kähler C-space G C /Q C., G/Q G C /Q C. G C /Q C Hermite. Kähler C-space Hermite, ( ). 6.5 F k1,...,k s (R n ) F k1,...,k s (C n ).

24 20 6 Hermite 6.3 Hermite 6.4 : 6.6 R-space, Hermite. 6.7 Hermite, 6.4., Hermite R-space. Hermite M, : (1) M 1 Lie g = g 1 g 0 g 1. (2) g l. (3) 1 Lie l = l 1 l 0 l 1. (4) (3), (1). R-space. 6.8 M = G 2 (R n+2 ) (= Q n (C)), : S n, S n 1 S 1 /Z 2, S n 2 S 2 /Z 2, S n 3 S 3 /Z 2,... (6.1)

25 21 [1] A. Arvanitoyeorgos, An introduction to Lie Groups and the Geometry of homogeneous Spaces. Student Mathematical Library, 22. American Mathematical Society, Providence, RI, [2] A. Besse, Einstein manifolds. Ergeb. Math., 10. Springer-Verlag, Berlin, [3] P. B. Eberlein, Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, [4] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, [5] S. Kaneyuki and H. Asano, Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112 (1988), [6] A. W. Knapp, Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, [7] A. L. Onishchik and È. B. Vinberg, Lie Groups and Lie Algebras, III. Structure of Lie groups and Lie algebras. Encyclopaedia of Mathematical Sciences, 41. Springer-Verlag, Berlin, [8] N. Tanaka, On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J. 8 (1979), [9] H. Tamaru, The local orbit types of symmetric spaces under the actions of the isotropy subgroups. Diff. Geom. Appl. 11 (1999), [10] H. Tamaru, On certain subalgebras of graded Lie algebras. Yokohama Math. J. 46 (1999),

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