Homogeneous groups

Let \delta_r be dilations on \mathbb{G}. We say that Lie group \mathbb{G} is a homogeneous group if:

(1)  It is a connected and simply connected nilpotent Lie group \mathbb{G} whose Lie algebra is endowed with a family of dilations \{\delta_r\}.

(2) The maps \exp\circ \delta_r \circ \exp^{-1} are group authomorphism of {\mathbb G}.

Since the exponential mapping is a global diffeomorphism, the dilations of the Lie algebra give rise to the corresponding dilations on the group.

So, roughly speaking: a homogeneous group is just {\mathbb R}^n but with a polynomial group law and a compatible system of dilations.

Now let us give some well-known examples for homogeneous groups.

Example 1. (Abelian groups) The Euclidean space \mathbb{R}^n is a homogeneous group with dilation given by the scalar multiplication.

Werner Heisenberg

Example 2. (Heisenberg groups) If n\in {\mathbb N}, the Heisenberg group \mathbb{H}^n is the group whose underlying manifold is \mathbb{C}^n \times \mathbb{R} and whose multiplication is given by


The Heisenberg group \mathbb{H}^n is a homogeneous group with dilations


gerald_b-_folland_2013stein3-2The consistent treatment of homogeneous groups goes back to G. B. Folland and E. M. Stein’s fundamental book: Hardy spaces on homogeneous groups. Princeton University Press, 1982.

⛳ But you can also have a look at a few pages of a more modern explanation of homogeneous groups.