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plained before. Denote by
b : g ’! n(g, D)
the identification of the vectorspace g with n(g, D) using the basis {X1, ..., Xdim g}.
The exponential and the logarithm of the groups G and N(G, D) will be denoted by
expG, logG and expN, logN respectively.
4 SUB-RIEMANNIAN LIE GROUPS 62
We shall consider the following N(G, D) atlas for G:
A = {Èx : Ux ‚" G ’! N(D, G) : x " G ,
Èx(y) = expN æ%b æ% logG(x-1y)
We shall study the differentiability properties of the transition functions. We want to
work in g. For this we have to transport (in a neighbourhood 0 " g) the interesting
operations, namely:
g
X · Y = logG (expG(X) expG(Y ))
n
X · Y = logN (expN (X) expN(Y ))
and to use instead of the chart Èx : Ux ‚" G ’! N(D, G), the chart ÆX : Ux ‚" G ’! g,
g
where x = expG X and ÆX(y) = (-X) · logG(y).
The transition function from ÆX to Æ0 is then the left translation by X, with respect
g
to the operation ·. We denote this translation by Lg . We want to know if this function
X
n
is Pansu derivable from (g, ·) to itself. See the difference: the function is certainly
g
derivable (or commutative smooth) in the sense of definition 4.9, for the operation ·,
n
but what about the derivability with respect to the operation ·? In order to answer we
shall use the following trick: a C" function f : N ’! N is Pansu derivable if and only if
it preserves the horizontal distribution and the derivative in each point is a morphism
from N to N.
The (classical) derivative of Lg moves the distribution Dn(Y ) = DLn (e) into
X Y
g
DLg Dn(Y ) ‚" TXg g. The horizontal distribution in X · Y , corresponding to the
X
·Y
g
n
group operation ·, is Dn(X · Y ). The difference between these two distributions is
measured by one of the linear transformations:
AX,Y : Txg g ’! Txg g
·Y ·Y
-1
-1
AX,Y = DLg (0) DLg (0) DLn (0) DLn (0) (4.3.10)
g
g
Y
Y
X ·Y
X ·Y
AX,Y : g ’! g
-1
-1
AX,Y = DLn (0) DLg (0) DLg (0) DLn (0) (4.3.11)
g
g
Y
Y
X ·Y
X ·Y
Let then J(G, D) be the Lie group generated by these transformations
J(G, D) = AX,Y : X, Y " g
It is then straightforward to see that the algebra j(G, D) of this group contains the
algebra generated by all the linear transformation with the form
ax = adG - adN (4.3.12)
x x
The necessary and sufficient condition for the group J (G, D) to be included in the
n
group End((g, ·), D) is that all the elements aX, to be in the algebra of the mentioned
linear group. But this is equivalent with one of the (equivalent) conditions:
4 SUB-RIEMANNIAN LIE GROUPS 63
(i) AdG is [·, ·]N morphism, for any x in a neighbourhood of the identity,
x
(ii) for any X, U, V " g we have the identity:
[[X, U]G, V ]N + [U, [X, V ]G]N = [X, [U, V ]N ]G (4.3.13)
We collect what we have found in the following theorem.
Theorem 4.16 Set AdG to be the adjoint representation of G and AdN the adjoint
representation ofN(G, D), seen as group of linear transformations on g (via the iden-
tification function b). The following are equivalent:
n
(a) J (G, D) ‚" End(g, ·),
n
(b) AdG ‚" End(g, ·),
(c) the relation (4.3.13) is true.
If (4.3.13) holds then AdGAdN is a group and its Lie algebra is the adjoint represen-
tation of the algebra g •" g with the bracket:
[(X, U), (Y, V )] = ([X, Y ]G, [X, V ]G + [U, Y ]G + [V, U]N ) (4.3.14)
Proof. The equivalence of (a), (b), (c) has been explained. Suppose now that (4.3.13)
is true. It is then straightforward to show that the space of all elements
a(X,Y ) = adG - adN
X Y
forms a Lie algebra with the linear commutator as bracket. Moreover, we have:
[a(X,U), a(Y,V )] = a[(X,U),(Y,V )]
This shows that AdGAdN is a group and the property of its algebra. In order to finish
the proof remark that a(X,X) = aX, defined at (4.3.12).
Corollary 4.17 The atlas A gives to G a C1 N(D, G) manifold structure if and only
if (4.3.13) is true and for any µ > 0, X, Y " g
-1 -1
´µ [X, ´µY ]N - ´µ [X, ´µY ]G = [X, Y ]N - [X, Y ]G (4.3.15)
4 SUB-RIEMANNIAN LIE GROUPS 64
Proof. The atlas A gives to G a C1 N(D, G) manifold structure if and only if
n
J (G, D) ‚" HL(g, ·) and any of its elements commute with dilatations ´µ. This
is equivalent with (4.3.13) and (4.3.15).
Remark 4.18 A Lie group is a manifold endowed with a smooth operation. In what
sense is then G a (sub-Riemannian) Lie group? We already have problems to assign an
atlas with smooth transition functions to G. The real meaning of the corollary 4.17 is [ Pobierz całość w formacie PDF ]