Lecture 16. Free Carnot groups

Posted on January 5, 2013 by Fabrice Baudoin

We introduce here the notion of Carnot group, which is the correct structure to understand the algebra of the iterated integrals of a path up to a given order. It is worth mentioning that these groups play a fundamental role in sub-Riemannian geometry as they appear as the tangent cones to sub-Riemannian manifolds.

Definition: A Carnot group of step (or depth) N is a simply connected Lie group \mathbb{G} whose Lie algebra can be written
\mathcal{V}_{1}\oplus...\oplus \mathcal{V}_{N},
with
\lbrack \mathcal{V}_{i},\mathcal{V}_{j}]=\mathcal{V}_{i+j}
and \mathcal{V}_{s}=0,\text{ for }s > N.

There are some basic examples of Carnot groups.

Example 1: The group \left( \mathbb{R}^d ,+ \right) is the only commutative Carnot group.

Example 2: (Heisenberg group) Consider the set \mathbb{H}_n =\mathbb{R}^{2n} \times \mathbb{R} endowed with the group law
(x,\alpha) \star (y, \beta)=\left( x+y, \alpha + \beta +\frac{1}{2} \omega (x,y) \right),
where \omega is the standard symplectic form on \mathbb{R}^{2n}, that is
\omega(x,y)= x^t \left( \begin{array}{ll} 0 & -\mathbf{I}_{n} \\ \mathbf{I}_{n} & ~~~0 \end{array} \right) y.
On \mathfrak{h}_n the Lie bracket is given by
[ (x,\alpha) , (y, \beta) ]=\left( 0, \omega (x,y) \right),
and it is easily seen that \mathfrak{h}_n=\mathcal{V}_1 \oplus \mathcal{V}_2, where \mathcal{V}_1 =\mathbb{R}^{2n} \times \{ 0 \} and \mathcal{V}_2= \{ 0 \} \times \mathbb{R}. Therefore \mathbb{H}_n is a Carnot group of depth 2.

The Carnot group \mathbb{G} is said to be free if \mathfrak{g} is isomorphic to the nilpotent free Lie algebra with d generators. In that case, \dim \mathcal{V}_{j} is the number of Hall words of length j in the free algebra with d generators. A combinatorial argument shows then that:
\dim \mathcal{V}_{j}= \frac{1}{j} \sum_{i \mid j} \mu (i) d^{\frac{j}{i}}, \text{ } j \leq N,
where \mu is the Möbius function. A consequence from this is that when N \rightarrow +\infty,
\dim \mathfrak{g} \sim \frac{d^N}{N}.
The free Carnot groups are the ones that will be the most relevant for us, so from now on, we will restrict our attention to them.

Let \mathbb{G} be a free Carnot group of step N. Notice that the vector space \mathcal{V}_{1}, which is called the basis of \mathbb{G}, Lie generates \mathfrak{g}, where \mathfrak{g} denotes the Lie algebra of \mathbb{G}. Since \mathbb{G} is step N nilpotent and simply connected, the exponential map is a diffeomorphism and the Baker-Campbell-Hausdorff formula therefore completely characterizes the group law of \mathbb{G} because for U,V \in \mathfrak{g},
\exp U \exp V = \exp \left( P (U,V) \right)
for some universal Lie polynomial P whose first terms are given by
P (U,V) = U+V+\frac{1}{2} [U,V] +\frac{1}{12} [[U,V],V]-\frac{1}{12}[[U,V],U]
-\frac{1}{48} [V,[U,[U,V]]]-\frac{1}{48} [U,[V,[U,V]]]+\cdots.
On \mathfrak{g} we can consider the family of linear operators \delta_{t}:\mathfrak{g} \rightarrow \mathfrak{g}, t \geq 0 which act by scalar multiplication t^{i} on \mathcal{V}_{i} . These operators are Lie algebra automorphisms due to the grading. The maps \delta_t induce Lie group automorphisms \Delta_t :\mathbb{G} \rightarrow \mathbb{G} which are called the canonical dilations of \mathbb{G}.

It is an interesting fact that every free Carnot group of step N is isomorphic to some \mathbb{R}^m endowed with a polynomial group law. Indeed, let X_1,\cdots,X_d be a basis of \mathcal{V}_{1}. From the Hall-Witt theorem we can construct a basis of \mathfrak{g} which is adapted to the grading
\mathfrak{g}=\mathcal{V}_{1}\oplus \cdots \oplus \mathcal{V}_{N},
and such that every element of this basis is an iterated bracket of the X_i‘s. Such basis, which can be made quite explicit, will be referred to as a Hall basis over X_1,\cdots,X_d. Let \mathcal{B} be such a basis. For X \in \mathfrak{g}, let [X]_\mathcal{B} be the coordinate vector of X in the basis \mathcal{B}. If we denote by m the dimension of \mathfrak{g}, we see that we can define a group law \star on \mathbb{R}^m by the requirement that for X,Y \in \mathfrak{g},
[X]_\mathcal{B} \star [Y]_\mathcal{B} =[ P_N(X,Y) ]_\mathcal{B}=[ \ln (e^X e^Y) ]_\mathcal{B}.
It is then clear that (\mathbb{R}^m, \star) is a Carnot group of step N whose Lie bracket is given by:
[ [X]_\mathcal{B} , [Y]_\mathcal{B}] =[ [X,Y] ]_\mathcal{B}.
Therefore, every free Carnot group of step N such that \dim \mathcal{V}_{1}=d is isomorphic to (\mathbb{R}^m, \star). Another representation of the free Carnot group of step N which is particularly adapted to rough paths theory is given in the framework of formal series. As before, let us denote by \mathbb{R}[[X_1, \cdots, X_d ]] the set formal series. Let us denote by \mathbb{R}_N[X_1,\cdots,X_d] the set of truncated series at order N, that is \mathbb{R}[[X_1, \cdots, X_d ]] quotiented by X_{i_1}\cdots X_{i_k}=0 if k \ge N+1. In this context, the free nilpotent Lie algebra of order N can be identified with the Lie algebra generated by X_1,\cdots,X_d inside \mathbb{R}_N[X_1,\cdots,X_d], where the bracket is of course given by the anticommutator. This representation of the free nilpotent Lie algebra of depth N shall be denoted by \mathfrak{g}_N(\mathbb{R}^d) in the sequel of the course. The free nilpotent group of step can then be represented as \mathbb{G}_N(\mathbb{R}^d)=\exp ( \mathfrak{g}_N(\mathbb{R}^d)) where the exponential map is the usual exponential of formal series.

We are now ready for the definition of the lift of a path in \mathbb{G}_N(\mathbb{R}^d).

Definition: Let x \in C^{1-var}([0,T],\mathbb{R}^d). The \mathbb{G}_N(\mathbb{R}^d) valued path
\sum_{k=0}^{N} \int_{\Delta^k [0,t]} dx^{\otimes k}, \quad 0 \le t \le T,
is called the lift of x in \mathbb{G}_N(\mathbb{R}^d) and will be denoted by S_N(x).

It is worth noticing that S_N(x) is indeed valued in \mathbb{G}_N(\mathbb{R}^d) because from the Chen’s expansion formula:
S_N(x)(t)=\exp \left( \sum_{k = 1}^N \sum_{I \in \{1,\cdots ,d\}^k}\Lambda_I (x)_{t} X_I \right),
where the notations have been introduced before. The multiplicativity property of the signature also immediately implies that for s \le t,
S_N(x)(t)=S_N(x)(s)\exp\left( \sum_{k = 1}^N \sum_{I \in \{1,\cdots ,d\}^k}\Lambda_I (x)_{s,t} X_I\right).

This entry was posted in Rough paths theory. Bookmark the permalink.

Leave a comment