---
tag: 3JVGZVVTB0
kind: concept
name: Maximal torus
msc: ["22E15", 22C05]
sources:
  - url: https://en.wikipedia.org/wiki/Maximal_torus
same_as:
  wikidata: Q1000120
  wikipedia: Maximal torus
---

## Summary

A maximal torus of a compact Lie group is a subgroup isomorphic to a torus $\mathbb{T}^k$ that is not contained in any larger such subgroup.
In a compact connected Lie group all maximal tori are conjugate, and every element lies in one of them.
Much of the theory of compact Lie groups is read off from a maximal torus and the action of the Weyl group on it.

## Explanations

### L1 human

A torus inside a group is a family of commuting rotations, like several independent clock hands.
A maximal torus is as large as such a family can be.
In the group $U(n)$ of unitary matrices, the diagonal matrices with entries on the unit circle form a maximal torus: there are $n$ independent angles.
Every unitary matrix can be diagonalised by a unitary change of basis, so every element of $U(n)$ lies in a conjugate of this torus.