---
tag: GA81VVQHEP
kind: concept
name: Linear algebraic group
msc: [20G15, 14L17]
specializes: [GH81SYAPPN]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_algebraic_group
same_as:
  wikidata: Q6553407
  wikipedia: Linear algebraic group
---

## Summary

A linear algebraic group is a group of invertible matrices defined by polynomial equations in the entries.
Equivalently, it is an [[algebraic group|GH81SYAPPN]] whose underlying scheme is affine.
Examples are the [[general linear group|0FCHE902D7]], the [[special linear group|0V9PMN0RBH]], the [[orthogonal groups|0TKYAEJ8N7]] and the [[symplectic groups|X12FS2JAP0]].
Their structure theory, through tori, Borel subgroups and root systems, is a large part of modern algebra and number theory.

## Explanations

### L1 human

Think of a group of matrices cut out by equations, such as the matrices $A$ with $A^{T} A = I$, or those with determinant $1$.
Such a group is at the same time a group and a geometric shape in the space of matrices.
Linear algebraic groups are the algebraic counterpart of matrix Lie groups, and they make sense over any field, including finite fields.
Over a finite field they give many of the finite simple groups.