**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 1Let 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,