From the Heisenberg group to Carnot groups

Transcription

1 From the Heisenberg group to Carnot groups p. 1/47 From the Heisenberg group to Carnot groups Irina Markina, University of Bergen, Norway Summer school Analysis - with Applications to Mathematical Physics Göttingen August 29 - September 2, 2011

2 From the Heisenberg group to Carnot groups p. 2/47 Simplest example of a sub-riemannian manifold Let M = R 3, q = (x,y,t), D = span{x,y} X = x 1 2 y t, Y = y x t. The metric g D is such that g D = ( ). (R 3,D,g D ) is the sub-riemannian manifold

3 From the Heisenberg group to Carnot groups p. 3/47 Example of a sub-riemannian manifold D is bracket generating, since [X,Y] = t := T and T q R 3 = span{x,y,t} for all q = (x,y,t) R 3.

4 Contact manifold It is contact manifold (M,ω), where M is a manifold of dimm = 2n+1 and ω (dω) n 0. The distribution D is the kernel of ω

5 From the Heisenberg group to Carnot groups p. 4/47 Contact manifold It is contact manifold (M,ω), where M is a manifold of dimm = 2n+1 and ω (dω) n 0. The distribution D is the kernel of ω and dual basis X = x 1 2 y t, Y = y x t, T = t dx, dy, ω = dt 1 2 xdy ydx D = span{ω} ω (dω) = dx dy dt.

6 From the Heisenberg group to Carnot groups p. 5/47 Geodesics D is strongly bracket generating, so all geodesics are normal. Write λ = ξdx+ηdy +θdt for λ TqM. H(q,λ) = 1 2 ( λ,x 2 +( λ,y 2 ) = 1 2 ((ξ 1 2 θy)2 +(η+ 1 2 θx)2 ). The Hamiltonian system and the initial conditions are ẋ = ξ 1 2 θy ẏ = η θx ṫ = 1 2 (ηx ξy)+ 1 4 θ(x2 +y 2 ) ξ = 1 2 ηθ 1 4 θx η = 1 2 ξθ 1 4 θy θ = 0, { x = y = t = 0, ξ = ξ 0, η = η 0, θ = θ 0.

7 From the Heisenberg group to Carnot groups p. 6/47 Geodesics We need projections of the bi-characteristic of H to R 3, so we reduce the Hamiltonian system to the system containing only the variables (x, y, t). {ẍ = θ0 ẏ ÿ = θ 0 ẋ, or ( ẍ ÿ ) = θ 0 ( )( ẋ ẏ Then multiplying the fist equation by y and the second by x we notice that the third equation is equivalent to the condition ṫ(s) = 1 2 (x(s)ẏ(s) y(s)ẋ(s)). ).

8 From the Heisenberg group to Carnot groups p. 7/47 Geodesics The solution is x(s) = ξ 0 θ 0 sin( θ 0 s) η 0 θ 0 (cos( θ 0 s) 1), y(s) = ξ 0 θ 0 (cos( θ 0 s) 1) η 0 θ 0 sin( θ 0 s), t(s) = (ξ2 0+η0) 2 2 θ 0 ( θ 2 0 s sin( θ 0 s)), if θ 0 0 and x(s) = ξ 0 s, y(s) = η 0 s, t(s) = 0, if θ 0 = 0.

9 Geodesics From the Heisenberg group to Carnot groups p. 8/47

10 From the Heisenberg group to Carnot groups p. 9/47 Geodesics ċ(s) = ẋ(s) x +ẏ(s) y +ṫ(s) t = ẋ(s) ( x 1 2 y(s) ) t + ẏ(s) ( y x(s) ) t + ( ṫ(s)+ 1 2ẋ(s)y(s) 1 2ẏ(s)x(s)) t = ẋ(s)x(c(s))+ẏ(s)y(c(s)) + ( ṫ(s)+ 1 2ẋ(s)y(s) 1 2ẏ(s)x(s)) T(c(s)). The curve is horizontal if and only if ṫ(s)+ 1 2ẋ(s)y(s) 1 2ẏ(s)x(s) = 0 or ω(ċ) = 0

11 From the Heisenberg group to Carnot groups p. 10/47 Geodesics It is known that in the Riemannian geometry if the neighborhood is small enough, then any two point can be connected by unique geodesic. It is not the case in sub-riemannian geometry! Take q 0 = (0,0,0) and q 1 = (x,y,t), x 2 +y 2 = 0. Then 0 = x 2 +y 2 = sin2 ( θ 0 ) θ 0 2 (ξ 2 0 +η 2 0) = θ 0 = 2πn, n N. t = 1 4 (ξ2 0 +η 2 0)

12 Heisenberg group = 1 x y t y x x 1 y 1 t y x x+x 1 y +y 1 t+t (xy 1 x 1 y) y+y x+x

13 From the Heisenberg group to Carnot groups p. 11/47 Heisenberg group = 1 x y t y x x 1 y 1 t y x x+x 1 y +y 1 t+t (xy 1 x 1 y) y+y x+x (x,y,t) (x 1,y 1,t 1 ) = (x+x 1,y +y 1,t+t (xy 1 x 1 y))

14 From the Heisenberg group to Carnot groups p. 12/47 Heisenberg group (R 3,+) (R 3, ) τ q = (x,y,t) (x 1,y 1,t 1 ) = (x+x 1,y+y 1,t+t (xy 1 x 1 y)). e = (0,0,0), τ 1 = ( x, y, t) So H 1 = (R 3, ) is non Abelian Lie group, dl τ = y 1 2 x 1. l τ (q) = τ q.

15 From the Heisenberg group to Carnot groups p. 13/47 dl τ = dl τ = dl τ = y 1 2 x y 1 2 x y 1 2 x = = [X,Y] = T Left invariant basis y x = = x 1 2 y t = X = x x t = Y = t = T

16 From the Heisenberg group to Carnot groups p. 14/47 Exponential map g 0 X tx exp G e τ exp: g G X τ

17 From the Heisenberg group to Carnot groups p. 15/47 Sophus Lie and Werner Heisenberg

18 Baker-Campbell-Hausdorff formula Coordinates of the first kind on H 1 H 1 q = (x,y,t) = exp(xx+yy +tt), xx+yy +tt h 1. [X,Y] = T

19 From the Heisenberg group to Carnot groups p. 16/47 Baker-Campbell-Hausdorff formula Coordinates of the first kind on H 1 H 1 q = (x,y,t) = exp(xx+yy +tt), xx+yy +tt h 1. [X,Y] = T Let V = xx +yy +tt, V 1 = x 1 X +y 1 Y +t 1 T τq = exp(v)exp(v 1 ) = exp(v +V [V,V 1]+...) = exp ( (x+x 1 )X +(y +y 1 )Y +(t+t 1 )T (xy 1 x 1 y)t ) = ( x+x 1,y +y 1,t+t (xy 1 x 1 y) ),

20 Norm and dilation (R 3,+, E ) (R 3,, H 1) q H 1 = ( (x 2 +y 2) 2 +t 2) 1/4.

21 From the Heisenberg group to Carnot groups p. 17/47 Norm and dilation (R 3,+, E ) (R 3,, H 1) ( (x q H 1 = 2 +y 2) 2 1/4. +t 2) δ s (q) E = (sx,sy,st) E = s q E. δ s (q) H 1 = (sx,sy,s 2 t) H 1 = s q H 1.

22 From the Heisenberg group to Carnot groups p. 18/47 Norm, dilation and distance (R 3,+, E ) (R 3,, H 1) ( (x q H 1 = 2 +y 2) 2 1/4. +t 2) δ s (q) E = (sx,sy,st) E = s q E. δ s (q) H 1 = (sx,sy,s 2 t) H 1 = s q H 1. d E (q 0,q 1 ) = q 0 q 1 E d H 1(q 0,q 1 ) = q 0 q 1 1 H 1

23 From the Heisenberg group to Carnot groups p. 19/47 Norm, dilation and distance (R 3,+, E ) (R 3,, H 1) q H 1 = ( (x 2 +y 2) 2 +t 2) 1/4. δ s (τ) E = (sx,sy,st) E = s τ E. δ s (τ) H 1 = (sx,sy,s 2 t) H 1 = s τ H 1. d E (q 0,q 1 ) = q 0 q 1 E Moreover d H 1(q 0,q 1 ) = q 0 q 1 1 H 1 d H 1(q 0,q 1 ) d c c (q 0,q 1 )

24 From the Heisenberg group to Carnot groups p. 20/47 Balls The Euclidean, Heisenberg, and Carnot-Carathéodory balls

25 Isoperimetric problem From the Heisenberg group to Carnot groups p. 21/47

26 Isoperimetric problem From the Heisenberg group to Carnot groups p. 22/47

27 Isoperimetric problem Given a curve c of the fixed length l and the straight line L (a part of Mediterranean sea-shore), put one end of c to L and determine the form of the figure that enclose the maximal area

28 From the Heisenberg group to Carnot groups p. 23/47 Isoperimetric problem Given a curve c of the fixed length l and the straight line L (a part of Mediterranean sea-shore), put one end of c to L and determine the form of the figure that enclose the maximal area c half circle length l(c) is fixed Area is maximal L Solution of the isoperimetric problem

29 From the Heisenberg group to Carnot groups p. 24/47 Dual isoperimetric problem dt = 1 2 (xdy ydx) is the area form. dt L = 0 for all straight line L passing through the origin. y c A(c) L 0 x Area A(c) is fixed Length l(c) is minimal Minimize the length l(c) of a curve c, under the condition that the enclosed area A(c) is constant A(c) = 1 2 c (xdy ydx), l(c) = c ẋ2 +ẏ 2 ds

30 From the Heisenberg group to Carnot groups p. 25/47 Isoperimetric problem Add the third coordinate t such that ṫ(s) = 1 2 ( ) x(s)ẏ(s) y(s)ẋ(s) for all s I. Then we lift the curve c R 2 up to a curve γ R 3

31 From the Heisenberg group to Carnot groups p. 26/47 Isoperimetric problem We want to preserve the length of the planar curve c. l(c) = l(γ) = ẋ2 +ẏ 2 ds It will be true if γ D γ, where D(x,y,t) = ker(ω) = ker ( dt 1 2 (xdy ydx)),

32 From the Heisenberg group to Carnot groups p. 27/47 Generalizations of the Heisenberg group At the level of groups (R 2+1, ) (R n+m, ) At the level of algebras (V 1 V 2, [X,Y] = T) (V 1 V 2,[V 1,V 1 ] V 2 ) or more general V 1 V 2... V m, [V 1,V 1 ] V 2 [V 1,V 2 ] = V 3... V m = [V 1,V m 1 ]

33 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R?

34 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R? What are complex numbers C?

35 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R? What are complex numbers C? It is a pair (a,b) of real numbers a,b R,

36 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R? What are complex numbers C? It is a pair (a,b) of real numbers a,b R, a = a conjugate to a real number,

37 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R? What are complex numbers C? It is a pair (a,b) of real numbers a,b R, a = a conjugate to a real number, (a,b) = (a, b) is conjugate to a complex number,

38 From the Heisenberg group to Carnot groups p. 28/47 Cayley-Dickson construction What are real numbers R? What are complex numbers C? It is a pair (a,b) of real numbers a,b R, a = a conjugate to a real number, (a,b) = (a, b) is conjugate to a complex number, multiplication (a,b)(c,d) = (ac db,a d+cb).

39 From the Heisenberg group to Carnot groups p. 29/47 Quaternions H It is a pair (a,b) of complex numbers a,b C,

40 From the Heisenberg group to Carnot groups p. 29/47 Quaternions H It is a pair (a,b) of complex numbers a,b C, (a,b) = (a, b) is conjugate to a quaternion,

41 From the Heisenberg group to Carnot groups p. 29/47 Quaternions H It is a pair (a,b) of complex numbers a,b C, (a,b) = (a, b) is conjugate to a quaternion, with the multiplication (a,b)(c,d) = (ac db,a d+cb).

42 From the Heisenberg group to Carnot groups p. 29/47 Quaternions H It is a pair (a,b) of complex numbers a,b C, (a,b) = (a, b) is conjugate to a quaternion, with the multiplication (a,b)(c,d) = (ac db,a d+cb). Sir William Rowan Hamilton,

43 From the Heisenberg group to Carnot groups p. 30/47 Octonions On December 26, 1843 John T. Graves wrote to W. Hamilton describing a new 8-dimensional algebra. In March of 1845, Arthur Cayley published a paper in the Philosophical Magazine where a brief description of the octonions contained.

44 From the Heisenberg group to Carnot groups p. 31/47 Octonions O It is a pair (a,b) of quaternion numbers a,b H,

45 From the Heisenberg group to Carnot groups p. 31/47 Octonions O It is a pair (a,b) of quaternion numbers a,b H, (a,b) = (a, b) is conjugate to an octonion,

46 From the Heisenberg group to Carnot groups p. 31/47 Octonions O It is a pair (a,b) of quaternion numbers a,b H, (a,b) = (a, b) is conjugate to an octonion, with the multiplication (a,b)(c,d) = (ac db,a d+cb).

47 From the Heisenberg group to Carnot groups p. 31/47 Octonions O It is a pair (a,b) of quaternion numbers a,b H, (a,b) = (a, b) is conjugate to an octonion, with the multiplication (a,b)(c,d) = (ac db,a d+cb)....

48 From the Heisenberg group to Carnot groups p. 32/47 Properties we loose R division algebra, associative, commutative, self conjugate,

49 From the Heisenberg group to Carnot groups p. 32/47 Properties we loose R division algebra, associative, commutative, self conjugate, C division algebra, associative, commutative,

50 From the Heisenberg group to Carnot groups p. 32/47 Properties we loose R division algebra, associative, commutative, self conjugate, C division algebra, associative, commutative, H division algebra, associative,

51 From the Heisenberg group to Carnot groups p. 32/47 Properties we loose R division algebra, associative, commutative, self conjugate, C division algebra, associative, commutative, H division algebra, associative, O division algebra,

52 From the Heisenberg group to Carnot groups p. 32/47 Properties we loose R division algebra, associative, commutative, self conjugate, C division algebra, associative, commutative, H division algebra, associative, O division algebra, if xy = 0, then x = 0 or y = 0.

53 From the Heisenberg group to Carnot groups p. 33/47 Unities R has only one 1 = (1,0), 1 2 = 1. Imaginary part has dimension 0.

54 From the Heisenberg group to Carnot groups p. 33/47 Unities R has only one 1 = (1,0), 1 2 = 1. Imaginary part has dimension 0. C: 1 = (1,0), 1 2 = 1 and i = (0,1), i 2 = 1. Imaginary part has dimension 1. z = a+ib = Rez +iimz, a,b R.

55 From the Heisenberg group to Carnot groups p. 34/47 Unities H: 1 = (1,0), 1 2 = 1 and i 1 = (i,0), i 2 = (0,1), i 3 = (0,i), i 2 1 = i 2 2 = i 2 3 = 1.

56 From the Heisenberg group to Carnot groups p. 34/47 Unities H: 1 = (1,0), 1 2 = 1 and i 1 = (i,0), i 2 = (0,1), i 3 = (0,i), i 2 1 = i 2 2 = i 2 3 = 1. q = a+i 1 b+i 2 c+i 3 d = Req +i 1 Im 1 q +i 2 Im 2 q +i 3 Im 3 q, a,b,c,d R.

57 From the Heisenberg group to Carnot groups p. 35/47 Memorial desk Figure 5: i 2 1 = i 2 2 = i 2 3 = 1.

58 From the Heisenberg group to Carnot groups p. 36/47 Unities O: 1 = (1,0), 1 2 = 1 and with j 1 = (i 1,0), j 2 = (i 2,0), j 3 = (i 3,0), j 4 = (0,1), j 5 = (0,i 1 ), j 6 = (0,i 2 ), j 7 = (0,i 3 ), j1 2 = j2 2 = j3 2 = j4 2 = j5 2 = j6 2 = j7 2 = 1. Imaginary part has dimension 7.

59 From the Heisenberg group to Carnot groups p. 37/47 Euclidean space Lie group (R n,+) with the identity element e = 0 The exponential map exp: R n T e R n is the identity map, so we identify elements of algebra with elements of the group. The algebra consists of constant vector fields X k = x k = x k. Since [ x k, x l] = 0, the Lie algebra of (R n,+) is V 1 V 2, V 1 = R n, V 2 = {0}.

60 From the Heisenberg group to Carnot groups p. 38/47 Heisenberg group H z = (x,y) C, t R, w = (a,b) C, s R (z,t) (w,s) = ( z +w,t+s+ 1 2 Im(z w) ) 1 2 Im(z w) = 1 ( ( 0 1 w Tr Jz ), J = )

61 From the Heisenberg group to Carnot groups p. 39/47 Quaternion H-type group q H, t 1,t 2,t 3 R, h H, s 1,s 2,s 3 R (q,t 1,t 2,t 3 ) (h,s 1,s 2,s 3 ) ( q +h, t 1 +s Im 1(q h), t 2 +s Im 2(q h), t 3 +s Im 3(q h) )

62 From the Heisenberg group to Carnot groups p. 40/47 Skew symmetric forms i 1 = , i 2 = , i 3 =

63 From the Heisenberg group to Carnot groups p. 41/47 Octonion H-type group p O, t = (t 1,...,t 7 ) R 7, r O, s = (s 1,...,s 7 ) R 7. (p,t) (r,s) = ( p+r,t+s+ 1 2 Im(p r) ), where Im(pr r) = ( Im 1 (p r),...,im 7 (p r) )

64 From the Heisenberg group to Carnot groups p. 42/47 Lie algebras of H-types Lie algebras are of two steps V 1 V 2, where V 1 = R, C, H or O and V 2 = {0}, R 1, R 3 or R 7 [V 1,V 1 ] = V 2 The horizontal distribution D is the left translation of V 1 The sub-riemannian metric g D is just Euclidean metric on D

65 From the Heisenberg group to Carnot groups p. 43/47 Two step Carnot groups are groups G = R α+β whose algebras g are nilpotent of step 2, graded, stratified: g = V 1 V 2, [V 1,V 1 ] V 2, [V 1,V 2 ] = {0}. If (v 1,v 2 ),(v 1,v 2 ) V 1 V 2, then [ (v1,v 2 ),(v 1,v 2) ] = (0,Ω(v 1,v 1)), where Ω: R α R α R β is a R β -valued skew symmetric form. If τ = (x,t), q = (x 1,t 1 ) G, then τq = (x,t)(x 1,t 1 ) := (x+x 1,t+t Ω(x,x 1)).

66 From the Heisenberg group to Carnot groups p. 44/47 Siegel upper half space Let (q 1,q 2 ) K, K = C 2, H 2 or O 2. The Siegel upper half space is U = {(q 1,q 2 ) K : 4Req 2 > q 1 2 }. Let [w,t] H-type group. Groups H act on U by [w,t].(q 1,q 2 ) (q 1 +w,q 2 + w w q 1 +i t), i t = dimv 2 l=1 i k t k.

67 From the Heisenberg group to Carnot groups p. 45/47 Siegel upper half space This mapping preserves the "height" function r(q 1,q 2 ) = 4Req 2 q 1 2. Action of H preserves the boundary U. H [w,t] (w, w 2 +it) U by action on (0,0) H(eisenberg) coordinates U (q 1,q 2 ) = (q 1,t,r), t k = Im k (q 2 ), r(q 1,q 2 ) = 4Re(q 2 ) q 1 2. If 4Re(q 2 ) = q 1 2, then U (q 1,q 2 ) = (q 1,t 1,...,t dimv2 ), r = r(q 1,q 2 ) = 0.

68 From the Heisenberg group to Carnot groups p. 46/47 Siegel upper half space q 2 r(q 1,q 2 ) H U [w,t] = (w, w 2 +it) U 0 q 1

69 From the Heisenberg group to Carnot groups p. 47/47 The end The end of the second lecture Thank you for your attention