---
tag: GH81SYAPPN
kind: concept
name: Algebraic group
msc: [14L10, 14L15, 20G15]
specializes: [TJ04VQP4NB]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_group
same_as:
  wikidata: Q1695400
  wikipedia: Algebraic group
---

## Summary

An algebraic group is a group whose elements are the points of an algebraic variety, with multiplication and inversion given by polynomial maps.
The two main families are the [[linear algebraic groups|GA81VVQHEP]], such as $\mathrm{GL}_n$, and the [[abelian varieties|R7W03TXRS9]], such as elliptic curves.
[[Chevalley's structure theorem|MZ6XDATQKR]] shows that every smooth connected algebraic group over a perfect field is built from one of each.
Algebraic groups play for algebraic geometry the role that Lie groups play for differential geometry.

## Explanations

### L1 human

The invertible $2 \times 2$ matrices form a group, and both the product and the inverse are given by formulas in the entries.
So the group is also a geometric object, the set where the determinant is nonzero.
An algebraic group is any group of this kind, where the elements form a variety and the group operations are polynomial maps.
Other examples are the numbers under addition, the nonzero numbers under multiplication, and the points of an elliptic curve.