---
tag: CXG6JZGMCK
kind: concept
name: Unimodular group
msc: ["22D05", "28C10"]
specializes: [AHV05ZBCYW]
sources:
  - url: https://en.wikipedia.org/wiki/Haar_measure
---

## Summary

A unimodular group is a [[locally compact group|AHV05ZBCYW]] whose left [[Haar measure|VY0HXD3GC4]] is also right-invariant, that is, whose [[modular function|63TX60RTRQ]] is identically $1$.
Abelian groups, compact groups, discrete groups, semisimple Lie groups and connected nilpotent Lie groups are unimodular.
Many results of harmonic analysis, such as the existence of lattices or the simplest form of trace formulas, require unimodularity.

## Explanations

### L1 human

In some groups, measuring sizes in a way that ignores left shifts automatically ignores right shifts as well.
These groups are called unimodular.
Commutative groups are the obvious example, because left and right shifts agree.
The group of affine maps $x \mapsto ax + b$ with $a > 0$ is the standard group that is not unimodular: its left and right Haar measures are $da \, db / a^2$ and $da \, db / a$.