---
tag: DK87YN458D
kind: concept
name: Nilpotent Lie group
msc: ["22E25", "22E15", "20F18"]
specializes: [9CNSVFQ801, 0Q7MMQHC1P]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_group
---

## Summary

A nilpotent Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[nilpotent group|0Q7MMQHC1P]].
So its lower central series reaches the trivial group after finitely many steps.
Equivalently, its Lie algebra is a nilpotent Lie algebra.
Every nilpotent Lie group is a [[solvable Lie group|9CNSVFQ801]], but not conversely.
For a connected and simply connected nilpotent Lie group, the exponential map from the Lie algebra is a diffeomorphism.
So such a group can be studied almost entirely through its Lie algebra.

## Explanations

### L1 human

A nilpotent Lie group is close to being commutative.
The commutators of its elements may be nontrivial, but commutators with commutators, repeated a fixed number of times, are always trivial.
The basic example is the Heisenberg group of upper triangular $3 \times 3$ matrices with $1$ on the diagonal.
There, any two elements commute up to a central element, and central elements commute with everything.
Groups of this kind model the position and momentum operators of quantum mechanics.