’1

lim δµ fx (xδµ y)

µ’0

is uniform with respect to x then f is classically derivable in x along the horizontal

directions and we have the connection between Pansu derivative and classical derivative:

’1

P f (x) = f (x) = DLf (x) (0)Df (x)DLx (0)

Conversely if f is classically derivable along the distribution, such that the quantity

’1

f (x)|D = DLf (x) (0)Df (x)DLx (0)|D

can be extended to an algebra morphism of N , an such that the convergence

1

(f (xδµ y) ’ f (x))

lim

µ’0 µ

(with y ∈ D) is uniform with respect to x then f is Pansu derivable.

De¬nition 4.24 A function f : N ’ N is noncommutative smooth if for any B ∈

V L(N ) there is a continuous function AB : N ’ V L(N ) such that:

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4 SUB-RIEMANNIAN LIE GROUPS

(a) for any x ∈ N the map

˜B

y ’ fx (y) = A(x)’1 fx (By)

is Pansu derivable in 0,

(b) the map

˜B

(x, y) ∈ N 2 ’ (f (x), Dfx (0)y) ∈ N 2

is continuous,

(c) the convergence

’1

Dfx (0)y = lim δµ fx (xδµ y)

µ’0

is uniform with respect to x.

The derivative of f in x is by de¬nition

˜I

Df (x) = AI (x)D fx (x)

Note that if f is invertible then we have the decomposition:

˜I

Df (x)v = AI (x) , Df (x)h = Dfx (x)

Moreover, if f is invertible and classically derivable then the conclusion of proposition

4.23 is still true, namely Df (x) = f (x), where the ”gradient” f (x) is given by

’1

DLf (x) (0)Df (x)DLx (0)

The derivative is called noncommutative because it does not commute with dilata-

tions.

Proposition 4.25 If f : N ’ N is commutative smooth then it is also noncommuta-

tive smooth.

If f, g : N ’ N are noncommutative smooth then so it is f —¦ g.

Proof. Because the noncommutative derivative splits the function in a horizontal

and a vertical part, it is su¬cient to prove that if f is commutative smooth and g is

noncommutative smooth such that D h g(x) = id then f —¦g is noncommutative smooth.

This is contained in the de¬niton of noncommutative smoothness.

The chain formula is true for noncommutative derivative. But obviously not for the

vertical and the horizontal parts.

With this de¬nition of smoothness the right translations in a Carnot group become

smooth. It is easy to show that

D Rx = Ad’1 , Dv Rx = Ad’1 , Dh Rx = id

x x

This smoothness de¬nition seems to be a good one for Carnot groups. It is not

good enough though, as the following example shows.

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4 SUB-RIEMANNIAN LIE GROUPS

4.6 Moment maps and the Heisenberg group

Let G = T k • N be the direct sum of the k dimensional torus T k with the Carnot

group N . The Lie algebra of G is g = Rk • N (recall that we use the same notation

for the Carnot group N and its Lie algebra).

The algebra g is Carnot, with respect to the dilatations

δµ (T, X) = (µT, δµ X)

The distribution is generated by Rk • V 1 and the nilpotentisation of G with respect to

this distribution is N = Rk • H.

The corollary 4.17 tells that G admits a N manifold structure. As the construction

of the noncommutative derivative is local in nature, we could reproduce it for N

manifolds. In this case that simply means: in reasonings of local nature one can work

on N instead of G.

Take now N = H(n) and a toric Hamiltonian action

(s1 , ..., sk ) ∈ T k ’ φ(s1 ,...,sk ) ∈ Sympl(R2n )

For any v ∈ Rk = Lie T k the in¬nitesimal generator is

d

ξv (x) = φexp tv (x)

dt |t=0

Let Hv be the Hamiltonian of t ’ φexp tv . The moment map associated to the Hamil-

tonian toric action is then

—

2n k

’

jφ : R R , j(x), v = Hv (x)

We shall associate to the toric action a function on T K — H(n).

˜

¦ : G = T k — H(n) ’ G , ¦(s, x, x) = (s, φs (x, x))

¯ ¯ (4.6.18)

˜

where φs is the lift of (φs , 0) (see theorem 3.3) and it has the expression:

(φs (x), x + Fs (x))

¯

Let us compute the ”gradient”

’1

¦(s, x¯) =

x DL¦(s,x,¯) (0) D¦(s, x, x) DL(s,x,¯) (0)

¯

x x

In matrix form with respect to the decomposition of the algebra

Rk • H(n) = Rk • R2n • R

we obtain the following expression:

«

1

0 0

¦(s, x, x) = ‚φs

¯ Dφs 0

‚s

a(s, x) 0 1

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4 SUB-RIEMANNIAN LIE GROUPS

where a(s, x) is the function:

‚F 1 ‚φ

’ ω(φ,

a(s, x) = )

‚s 2 ‚s

We can improve the expression of the function a if we use the fact that the toric action

is Hamiltonian. Indeed, by direct computation one can show that

‚a ‚

(s, x) = ’ (jφ (φs (x)))

‚x ‚x

This is equivalent with

a(s, x) = »(s) ’ jφ (φs (x))

Easy computation shows that End(Rk • H(n)) is made by all matrices with the

form:

«

A00 — —

B0

0 B 0 , A ∈ GL(Rk ) , k k

∈ HL(H(n)) , a ∈ R = Lie T

bc

abc

It follows that ¦ ∈ End(Rk • H(n)) and we already saw this in a particular form

in proposition 3.6. Therefore ¦ is not even noncommutative smooth in the sense of

de¬nition 4.24.

Suppose that the toric action is by di¬eomorphisms with compact support. Then

»(s) can be taken equal to 0 hence a(x, s) = ’jφ (x).

Proposition 4.26 The map ¦ transforms the distribution

1

¯ ¯

DN (s, x, x) = (S, Y, Y ) : ’ ω(x, Y ) + Y = 0

¯

2

in the distribution

1

¯ ¯

Dφ (s, x, x) = ’

¯ (S, Y, Y ) : jφ (x), S ω(x, Y ) + Y = 0

2

Proof. By direct computation.

4.7 General noncommutative smoothness

The introduction of noncommutative derivative makes right translations in Carnot

groups ”smooth”. We shall apply the same strategy for having a manifold structure

compatible with the group operation, in the case of a Lie group G with a left invariant

distribution D.

’1

The transition functions of the chosen atlas have the form Lg , with X ∈ g.

X

These transition functions forms a group. Note that a natural generalization is the case

where the transition functions forms a groupoid. This generalisation is not meaningless

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4 SUB-RIEMANNIAN LIE GROUPS

since it leads to the introduction of intrinsic SR manifolds, if we identify a manifold

with the groupoid of transition functions of an atlas.

We want this family of transition functions to be ”smooth”. We shall proceed then

as in the motivating example.

The group G is identi¬ed with the group of left translations LG acting on the Carnot

n

group (g, ·). These transformation are not smooth, they preserve another distribution

than the N left invariant distribution on G.

We shall give the following ”upgraded” variant of the de¬nition 4.24, concerning

the noncommutative derivative.

n

We shall denote by N the set g endowed with the nilpotent multiplication ·. The

same notation will be used for the Lie algebra (g, [·, ·]N ). We ¬x on N an Euclidean

norm, given, for example, by declaring orthonormal a basis constructed from multi-

brackets from a basis of D. The Euclidean norm will be denoted by · .

We introduce the group of linear transformations Σ(G, D) generated by all trans-

formations DLg (0), DRX (0), HL(N ). In particular δµ ∈ Σ(G, D) for any µ > 0. From

g

X

the formula:

DLN (0) = lim δµ —¦ DLgµ X (0) —¦ δµ

’1

X δ

µ’0

we get that DLN (0) ∈ Σ(G, D). In the same way we obtain DRX (0) ∈ Σ(G, D).

N

X

For any F ∈ Σ(G, D) we de¬ne the group N (F ) to be the Carnot group N with

the bracket:

[X, Y ]F = F [F ’1 X, F ’1 Y ]N

N

and with the dilatations

δµ = F δµ F ’1

F

This group is endowed with the Carnot-Carath´odory distance generated by the Eu-

e

N (F ) = F D.

clidean norm and the distribution D

We can see the class {N (F ) : F ∈ Σ(G, D)} in two ways, as an orbit. First, it is

the orbit

¯

Σ(G, D) = (F δµ F ’1 , F adN’1 X F ’1 ) : F ∈ Σ(G, D)

F

in the space GL(N ) — gl(N, gl(N )), under the action of Σ(G, D)

F.(A, B) = (F AF ’1 , F (B)) , F (B)(Z) = F B(F ’1 Z)F ’1

Second, it is the right coset Σ(G, D)/HL(N ). Indeed, if N (F ) = N (P ) then F ’1 P ∈

HL(N ).

De¬nition 4.27 A function f : N ’ N is (G, D) noncommutative derivable in x ∈ N

if for any F ∈ Σ(G, D) there exists P ∈ Σ(G, D) such that f : N (F ) ’ N (P ) is Pansu

derivable in x. The derivative of f in x is by de¬nition:

Df (x)(F, P ) = P ’1 —¦ DP ansu f (x) —¦ F

The function f is noncommutative smooth if