1. Stratified nilpotent groups
aka Carnot groups.
1.1. Examples: Heisenberg groups
It was known to physicists under the name Weyl’s group. It was re-christened Heisenberg group by Elias Stein and his school of harmonic analysis.
is a multiplication on . The Lie algebra with , all other brackets vanishing. The bracket on is the symplectic form . Spelling ,
We assign a formal degree 1 to and 2 to and define the nonisotropic Lie algebra dilation
The nonisotropic group dilation is given by the same formula. Its Jacobian equals . The exponent is called homogeneous dimension, it will play the role of dimension in Euclidean analysis.
1.2. Carnot groups
A stratified nilpotent Lie algebra of step takes the form with , , and .
A Lie group is a Carnot group of step if the corresponding Lie algebra is stratified nilpotent of step .
Define . These are Lie algebra automorphisms. Since is nilpotent, the group exponential map is an analytic diffeomorphism. So are group automorphisms, the nonisotropic group dilations.
Fix an inner product on that makes ‘s orthogonal. The nonisotropic gauge on is
It is homogeneous of degree one under dilations . This is not a norm in the usual sense. Never mind. Using , we get a nonisotropic gauge on , which is homogeneous under .
The Baker-Campbell-Hausdorff formula expresses the multiplication rule of in exponential coordinates,
The full series can be found in books. For nilpotent groups, it is a finite sum. Exercise: compute the multiplication for .
1.3. The sub-Laplacian
Let be an orthonormal basis of . They represent left-invariant vectorfields on : where is left translation by , . Define
It depends on the choice of orthonormal basis. Thanks to Hörmander’s theorem, is hypoelliptic. It is left-invariant and homogeneous of degree 2 under dilations.
1.4. The -map
For step 2 groups, the following map is useful. Assume that the chosen inner product on makes and orthogonal. is defined by
For convenience, fix an orthonormal basis of .
Proposition 1 Let be a step 2 Carnot group. Then
1.5. Groups of Métivier type
These are the step 2 Carnot groups for which is nondegenerate,
1.6. Groups of Heisenberg type
These are the step 2 Carnot groups for which is isometric,
They were introduced by Aroldo Kaplan in research concerning hypoellipticity. Although it is a much smaller class that Métivier’s, it turns out to be quite rich. It contains quaternionic Heisenberg groups and many variants. It played a historical role in progress in understanding hypoellipticity.
For such groups, the formula for simplifies,