**1. Liouville’s theorem **

Let be a conformal map on an open domain of , . Observe that is conformal if and only if at every point , belongs to the conformal group such that . In other words, is an automorphism of the constant -structure, .

Infinitesimal conformal transformations are of the form

where the matrix belongs to the Lie algebra

** 1.1. First step : prolongation and infinitesimal transformations **

The prolongation is of the form , it is isomorphic, as a Lie algebra, to . This is a classical computation. This is a Liouville theorem for vectorfields.

** 1.2. Second step : conformal transformations **

Assume that . Let an infinitesimal conformal transformations. So is , but, a priori, is only . Mollify it. Since the equations are linear with constant coefficients, the mollified vectorfield is still conformal, thus belongs to . As a limit of such, . Whence a group homomorphism .

One show that this homomorphism is injective. Indeed, assume that and for all . Then maps a constant vectorfield to itself, its differential equals identity, it is a translation, therefore it equals identity.

On the Möbius group , is identity, thus showing that .

**2. Tanaka prolongation theory **

Let be a Lie group. Assume leaves a vectorsubspace invariant. A -structure subsumes a plane distribution. The Singer-Sternberg prolongation works if the -structure is integrable, which often forces the distribution to be involutive, and we do not want this.

Tanaka replaces constant -structures by different models: left-invariant -structures on stratified Lie groups.

Definition 1A model space (for Tanaka) is a stratified Lie algebrasuch that for all ,

The corresponding Lie group comes equipped with a horizontal distribution . Contact mappings (mappings that preserve ) can be viewed as automorphisms of a geometric structure.

Definition 2An infinitesimal automorphism is a vectorfield whose local flow consists of contact transformations.

Remark 1is an infinitesimal contact transformation if and only if .

** 2.1. The full prolongation **

We first define the full prolongation, i.e. Lie algebra relevant to the determination of all infinitesimal contact transformations.

Definition 3, where is the space of degree derivations, and for positive ,

Note that if , then .

Definition 4The Lie bracket structure is defined as follows. For , and , set and inductively

Definition 5Replacing by a subalgebra , one gets .

** 2.2. Finiteness criterion **

Definition 6Define

Note can be viewed as a subalgebra of , and thus admits a prolongation in the sense of Singer and Sternberg.

Theorem 7 (Tanaka)is finite dimensional if and only if is of finite type in the sense of Singer and Sternberg.

** 2.3. Examples **

Example 1Let be Heisenberg group. For the full prolongation, contains rank one matrices, so the full prolongation is infinite dimensional.

Example 2Let be Heisenberg group. Let whose restriction to belongs to . Then , so the prolongation is finite dimensional.

This implies that infinitesimal conformal contact transformations form a finite dimensional Lie algebra, but we have not proved it yet. The prolongation can be computed to be

Example 3Let be the group of unipotent upper triangular -matrices. Then the full prolongation isWhereas vanishes.

** 2.4. Prolongation of contact vectorfields **

We are aiming at relating the Lie algebra of infinitesimal contact transformations to the full prolongation.

Definition 8View a vectorfield on as a -valued function on , .

Note that .

If is a contact vectorfield and is horizontal and left-invariant, then is horizontal again, and

This implies that the right hand side is horizontal, i.e., for all ,

This set of equations characterizes infinitesimal contact transformations.

Example 4Let be the Heisenberg group. Then is a contact vectorfield if and only if and .

Indeed, in this example,

Iterating the equation above gives, for all , and left-invariant ,

Proposition 9Let be a vectorfield. The expressiondefines a degree derivation of . Therefore, it defines a map .

Example 5Heisenberg again. Then

The contact equations are expressible in terms of , , , …. which are higher and higher derivatives. This provides us with a map from infinitesimal contact transformations to the full prolongation. One shows that this map is an isomorphism.

Alessandro Ottazzi’s own notes of this minicourse can be found here.