Let be dilations on . We say that Lie group is a homogeneous group if:
(1) It is a connected and simply connected nilpotent Lie group whose Lie algebra is endowed with a family of dilations .
(2) The maps are group authomorphism of .
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 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 is a homogeneous group with dilation given by the scalar multiplication.
Example 2. (Heisenberg groups) If , the Heisenberg group is the group whose underlying manifold is and whose multiplication is given by
The Heisenberg group is a homogeneous group with dilations
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.