# 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.

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

$(z_1,...,z_n,t)(z_1',...,z_n',t')=(z_1+z_1',...,z_n+z_n',t+t'+2Im\sum_{k=1}^{n}z_k\overline{z}'_k).$

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

$\delta_r(z_1,...,z_n,t)=(rz_1,...,rz_n,r^2t).$

The 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.