is a Lie bracket (bilinear, antisymmetric and it satis¬es the Jacobi identity). Therefore

at the limit µ ’ 0 we get a Lie bracket. Moreover, it is straightforward to see from the

de¬nition of [x, y]n that δµ is an algebra isomorphism. We conclude that (g, [·, ·]n ) is

the Lie algebra of a Carnot group with dilatations δµ .

Proposition 4.3 Let {X1 , ..., Xdim g} be a basis of g constructed from a basis {X1 , ..., Xp }

of D. Set

Wj = span {Xi : l(Xi ) = j}

Then the spaces Wj provides a direct sum decomposition (4.1.2). Moreover the identi-

ˆ

¬cation Xi = Xi gives a Lie algebra isomorphism between (g, [·, ·]n ) and n(g, D).

In the basis {X1 , ..., Xdim g} the Lie bracket on g looks like this:

Proof.

[Xi , Xj ] = Cijk Xk

where cijk = 0 if l(Xi ) + l(Xj ) < l(Xk ). From here the ¬rst part of the proposition

is straightforward. The expression of the Lie bracket generated by the decomposition

(4.1.2) is obtained from (4.1.4)). We have

[Xi , Xj ]n = »ijk Cijk Xk

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4 SUB-RIEMANNIAN LIE GROUPS

where »ikj = 1 if l(Xi ) + l(Xj ) = l(Xk ) and 0 otherwise. The Lie algebra isomorphism

follows from the expression of the Lie bracket on n(g, D):

ˆˆ

[Xi , Xj ] = [Xi , Xj ] + V l(Xi )+l(Xj )’1

In conclusion the expression of the nilpotent Lie bracket depends on the choice of

basis B trough the transport of the dilatations group from n(g, D) to g.

Let N (G, D) be the simply connected Lie group with Lie algebra n(g, D). As

previously, we identify N (G, D) with n(g, D) by the exponential map.

4.2 Commutative smoothness for uniform groups

This section is just a sketch and certainly needs to be improved. It is inspired from

Bella¨

±che [3], last section. I think that by improving the notions explained here and

by making axioms from the conclusions of lemma 4.13 and Ball-Box theorem 4.19,

then it is possible to avoid the use of the notion of Lie group. For example, one could

use consequences of proposition 4.7 in order to construct a general manifold structure,

noncommutative derivative, moment map, as in sections 4.3, 4.5, 4.7.

We start with the following setting: G is a topological group endowed with an uni-

formity such that the operation is uniformly continuous. More speci¬cally, we introduce

¬rst the double of G, as the group G(2) = G — G with operation

(x, u)(y, v) = (xy, y ’1 uyv)

The operation on the group G, seen as the function

op : G(2) ’ G , op(x, y) = xy

is a group morphism. Also the inclusions:

i : G ’ G(2) , i (x) = (x, e)

i” : G ’ G(2) , i”(x) = (x, x’1 )

are group morphisms.

De¬nition 4.4 G is an uniform group if we have two uniformity structures, on G and

G2 , such that op, i , i” are uniformly continuous.

A local action of a uniform group G on a uniform pointed space (X, x0 ) is a function

ˆ ˆ

φ ∈ W ∈ V(e) ’ φ : Uφ ∈ V(x0 ) ’ Vφ ∈ V(x0 ) such that the map (φ, x) ’ φ(x) is

uniformly continuous from G — X (with product uniformity) to to X. Moreover, the

action has the property: for any φ, ψ ∈ G there is D ∈ V(x0 ) such that for any x ∈ D

ˆ ˆ

ˆˆ ˆˆ

φψ ’1 (x) and φ(ψ ’1 (x)) make sense and φψ ’1 (x) = φ(ψ ’1 (x)).

Finally, a local group is an uniform space G with an operation de¬ned in a neigh-

bourhood of (e, e) ‚ G — G which satis¬es the uniform group axioms locally.

Remark that a local group acts locally at left (and also by conjugation) on itself.

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4 SUB-RIEMANNIAN LIE GROUPS

De¬nition 4.5 A conical local uniform group N is a local group with a local action of

(0, +∞) by morphisms δµ such that limµ’0 δµ x = e for any x in a neighbourhood of

the neutral element e.

We shall make the following hypotheses on the local uniform group G: there is a

local action of (0, +∞) (denoted by δ), on (G, e) such that

H0. δµ (e) = e and limµ’0 δµ x = e uniformly with respect to x.

H1. the limit

’1

β(x, y) = lim δµ ((δµ x)(δµ y))

µ’0

is well de¬ned in a neighbourhood of e and the limit is uniform.

H2. the limit

x’1 = lim δµ (δµ x)’1

’1

ˆ

µ’0

is well de¬ned in a neighbourhood of e and the limit is uniform.

Proposition 4.6 Under the hypotheses H0, H1, H2 (G, β) is a conical local uniform

group.

Proof. All the uniformity assumptions permit to change at will the order of taking

limits. We shall not insist on this further and we shall concentrate on the algebraic

aspects.

We have to prove the associativity, existence of neutral element, existence of inverse

and the property of being conical. The proof is straightforward. For the associativity

β(x, β(y, z)) = β(β(x, y), z) we compute:

’1

β(x, β(y, z)) = lim δµ (δµ x)δµ/· ((δ· y)(δ· z))

µ’0,·’0

We take µ = · and we get

= β(x, β(y, z)) = lim {(δµ x)(δµ y)(δµ z)}

µ’0

In the same way:

’1

β(β(x, y), z) = lim δµ (δµ/· x) ((δ· x)(δ· y)) (δµ z)

µ’0,·’0

and again taking µ = · we obtain

β(β(x, y), z) = lim {(δµ x)(δµ y)(δµ z)}

µ’0

The neutral element is e, from H0 (¬rst part): β(x, e) = β(e, x) = x. The inverse of

x is x’1 , by a similar argument:

ˆ

β(x, x’1 ) = δµ (δµ x) δµ/· (δ· x)’1

’1

ˆ lim

µ’0,·’0

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4 SUB-RIEMANNIAN LIE GROUPS

and taking µ = · we obtain

β(x, x’1 ) = lim δµ (δµ x)(δµ x)’1

’1 ’1

ˆ = lim δµ (e) = e

µ’0 µ’0

Finally, β has the property:

β(δ· x, δ· y) = δ· β(x, y)

which comes from the de¬nition of β and commutativity of multiplication in (0, +∞).

This proves that (G, β) is conical.

We arrive at a natural realization of the tangent space to the neutral element. Let

us denote by [f, g] = f —¦ g —¦ f ’1 —¦ g’1 the commutator of two transformations. For

the group we shall denote by LG y = xy the left translation and by LN y = β(x, y).

x x

The preceding proposition tells us that (G, β) acts locally by left translations on G.

Proposition 4.7 We have the equality

’1

lim δ» [δµ , LG » x)’1 ]δ» = LN (4.2.5)

x

(δ

»¤µ’0

and the limit is uniform with respect to x.

Proof. We shall use the equality:

lim δµ (β((δµ x), δµ y))) = lim δµ β(ˆ’1 , δµ (β(x, y)))

’1 ’1

x (4.2.6)

µ’0 µ’0

which is a consequence of H0 and the previous proposition. The left hand side of this

equality equals β(x, y). We shall look now at the right hand side (RHS) of the equality.

The uniformity assumptions and some computations lead us to the following equality,

under the constraint 0 < max {µ, », ·} < µ:

δµ/µ δµ/» (δ» x)’1 δµ/(µ·) ((δ· x)(δ· y))

RHS = lim

(µ,µ,»,·)’0

The mentioned constraint forces all expressions to make sense. Take now µ = » = · < µ

and get

’1

RHS = lim δ» δµ (δ» x)’1 δµ ((δ» x)(δ» y))

’1

µ,»’0

as desired.

An easy corollary is that

’1

lim [LG » x)’1 , δ» ] = LN

x

(δ

»’0

De¬nition 4.8 The group V Te G formed by all transformations LN is called the virtual

x

tangent space at e to G.

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4 SUB-RIEMANNIAN LIE GROUPS

The virtual tangent space V Tx G at x ∈ G to G is obtained by translating the group

operation and the dilatations from e to x. This means: de¬ne a new operation on G

by

x

y · z = yx’1 z

The group G with this operation is isomorphic to G with old operation and the left

translation LG y = xy is the isomorphism. The neutral element is x. Introduce also

x

the dilatations based at x by

δµ y = xδµ (x’1 y)

x

x

Then Gx = (G, ·) with the group of dilatations δµ satisfy the axioms Ho, H1, H2.

x

De¬ne then the virtual tangent space V Tx G to be: V Tx G = V Tx Gx . A short

computation using proposition 4.7 shows that

LN,x = Lx LN’1 y Lx : y ∈ Ux ∈ V(X)

V Tx G = y x

where

’1,x x

LN,x = lim δ» [δµ , LG » x)x,’1 ]δ»

x

y (δ

»¤µ’0

We shall introduce the notion of commutative smoothness, which contains a deriva-

tive resembling with Pansu derivative. Independently, the author introduced a general

”topological” derivative in [4] (there are a lot of typographical errors in this reference,

but the ideas behind are quite nice in my opinion). Lack of knowledge stopped the

development of this notion until recently, when I have learned about Carnot groups

and Pansu derivative. As it will be seen further, interesting information will be found

when noncommutative smoothness is considered.

De¬nition 4.9 A function f : G1 ’ G2 is commutative smooth, where G1 , G2 are two

groups satisfying H0, H1, H2, if the application

(2) (2)

(x, u) ∈ G1 ’ (f (x), Df (x)u) ∈ G2

exists and it is continuous, where

Df (x)u = lim δµ f (x)’1 f (xδµ u)

’1

µ’0

and the convergence is uniform with respect to (x, u).

For example the left translations Lx are commutative smooth and the derivative

equals identity. If we want to see how the derivative moves the virtual tangent spaces

we have to give a de¬nition.

De¬nition 4.10 Le f : G ’ G be a commutative smooth function. The virtual tangent

to f is de¬ned by: