Lecture 16. Free Carnot groups

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},
\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).

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