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 1 A model space (for Tanaka) is a stratified Lie algebra
such 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 2 An infinitesimal automorphism is a vectorfield whose local flow consists of contact transformations.
Remark 1 is 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 4 The Lie bracket structure is defined as follows. For , and , set and inductively
Definition 5 Replacing by a subalgebra , one gets .
2.2. Finiteness criterion
Definition 6 Define
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.
Example 1 Let be Heisenberg group. For the full prolongation, contains rank one matrices, so the full prolongation is infinite dimensional.
Example 2 Let 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 3 Let be the group of unipotent upper triangular -matrices. Then the full prolongation is
2.4. Prolongation of contact vectorfields
We are aiming at relating the Lie algebra of infinitesimal contact transformations to the full prolongation.
Definition 8 View 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 4 Let 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 9 Let be a vectorfield. The expression
defines a degree derivation of . Therefore, it defines a map .
Example 5 Heisenberg 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.