A Lie algebra is said to be graded when it is endowed with a vector space decomposition (where all but finitely many of the ‘s are ):
such that .
Then, a Lie group is said to be graded when it is a connected simply connected Lie group whose Lie algebra is graded.
The condition that the group is connected and simply connected may look scary but it is only technical, to ensure that the exponential mapping is a global diffeomorphism between the group and its Lie algebra.
The classical examples of graded Lie groups and algebras are the following.
Example. (Stratified groups) The stratified group are the graded groups with the extra conditions that generates the whole of the Lie algebra by taking iterated commutators. The vector fields from part (2) in the description of stratified groups give the space
So, in particular, the Abelian group is graded: its Lie algebra is trivially graded, i.e. .
Also, the Heisenberg group is graded: its Lie algebra can be decomposed as
where and .
You can read some small text here with more explanations and examples.