---
tag: 66FQFAJNS4
kind: concept
name: Algebraic torus
msc: [20G15, 14L10]
specializes: [GA81VVQHEP]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_torus
same_as:
  wikidata: Q4724022
  wikipedia: Algebraic torus
---

## Summary

An algebraic torus is a [[linear algebraic group|GA81VVQHEP]] that becomes a power of the multiplicative group $\mathbb{G}_m$ over an algebraic closure of the base field.
Over an algebraically closed field it is a group of invertible diagonal matrices.
Maximal tori control the structure of [[reductive groups|2XS84PWE3S]] through their characters and roots.
Tori also give [[toric varieties|YWB981MEF2]].

## Explanations

### L1 human

The simplest torus is the set of nonzero numbers under multiplication.
A torus of dimension $n$ is the set of $n$-tuples of nonzero numbers, multiplied coordinatewise, or equivalently the invertible diagonal $n \times n$ matrices.
Over the real numbers there are also twisted tori, such as the circle $x^2 + y^2 = 1$ with the multiplication of complex numbers $x + iy$.
Over $\C$ the circle becomes the same as $\C^{*}$ through $(x, y) \mapsto x + iy$.