# Nilpotent groups

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. linkdownload this book

Roughly speaking, a Lie group ${\mathbb G}$ 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 ${\mathbb R}^n$ 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 ${\mathfrak G}$ to the group ${\mathbb G}$ 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 ${\mathbb G}$ with ${\mathbb R}^n$ (for some $n$) so that the group law is polynomial in coordinate functions;
• the Haar measure on ${\mathbb G}$ is just the Lebesgue measure.

You can quickly check these and other properties of nilpotent Lie groups here.