---
tag: 9CNSVFQ801
kind: concept
name: Solvable Lie group
msc: ["22E25", "22E15", "20F16"]
specializes: [7JBRGCVJRB, 0FXJ7YFMMX]
sources:
  - url: https://en.wikipedia.org/wiki/Solvmanifold
---

## Summary

A solvable Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[solvable group|0FXJ7YFMMX]]: its derived series reaches the trivial group after finitely many steps.
Equivalently, its Lie algebra is a solvable Lie algebra.
Solvable Lie groups are the counterpart of semisimple ones.
By the Levi decomposition, every connected Lie group is built from a solvable normal subgroup and a semisimple part.
Typical examples are groups of invertible upper triangular matrices.

## Explanations

### L1 human

A solvable Lie group is one that can be taken apart, in finitely many steps, into commutative layers.
Take the commutators of the group, then the commutators of those, and so on: in a solvable group this process ends with the identity alone.
The group of maps $x \mapsto a x + b$ of the real line with $a > 0$ is an example.
Its commutators are the translations $x \mapsto x + b$, and these commute with each other, so the process stops after two steps.