There are many good sources of information on nilpotent Lie groups. On these pages we follow extracts from the following open access book:
Fischer V., Ruzhansky M., Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp. link, download this book
Roughly speaking, a Lie group is called nilpotent if only finitely many iterated commutators of vector fields on the group are non-zero. The step of the group is the largest number of vector fields for which the iterated commutator is non-zero.
For example, the usual Euclidean group is nilpotent of step 1 (all commutators are zero). The Heisenberg group is a nilpotent Lie group of step 2. The Engel group and the Dynin-Folland group are nilpotent of step 3, and the Cartan group is nilpotent of step 4, etc.
The condition that a group is nilpotent has a number of consequences (modulo small technicalities):
- the exponential mapping from the Lie algebra of to the group is a global diffeomorphism;
- it follows from the Poincaré–Birkhoff–Witt theorem that the group law is polynomial. Consequently, we can always identify a nilpotent Lie group with (for some ) so that the group law is polynomial in coordinate functions;
- the Haar measure on is just the Lebesgue measure.
You can quickly check these and other properties of nilpotent Lie groups here.