# General linear group over a field as an algebraic group

From Groupprops

## Contents

## Definition

Suppose is a field and is a natural number. The general linear group is defined as the group of all invertible matrices over . This has the natural structure of an algebraic group as described below:

- The algebraic variety structures arises as follows: it is a Zariski-open subset of , defined by the non-vanishing of the determinant polynomial.
- The group structure is by matrix multiplication. Both the matrix multiplication and the inverse map are given
*globally*by rational functions in the coordinates, and hence are morphisms of algebraic varieties.

## Particular cases

### General linear group of degree one as an algebraic group

For , the general linear group is the subset of the field . Explicitly:

Item | Description | Comment |
---|---|---|

Definition as Zariski-open subset of | ||

Explicit formula for group multiplication | Group multiplication is field multiplication. | |

Explicit formula for inverse map in group | Group inverse is field inverse. |

### General linear group of degree two as an algebraic group

Here, we write a matrix as a bunch of four coordinates . Then:

Item | Description | Comment |
---|---|---|

Definition as Zariski-open subset of | Nonvanishing determinant condition | |

Explicit formula for group multiplication | Group multiplication is matrix multiplication. | |

Explicit formula for inverse map in group | Group inverse is matrix inverse. Note that denominators do not vanish, and this is precisely because the determinant is nonvanishing on this open subset. |