Nilpotent groups

Cover-Fischer-RuzhanskyThere 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 {\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.