# Dimension of an algebraic group

From Groupprops

## Definition

The **dimension** of an algebraic group over a field is defined in the following equivalent ways:

No. | Shorthand | Definition |
---|---|---|

1 | algebraic variety | its dimension as an algebraic variety over the field over which it is defined |

2 | formal group law | the dimension of the formal group law associated with the algebraic group |

3 | Lie algebra dimension | the dimension (as a vector space over the field) of the Lie algebra of the algebraic group. |

## Facts

- The dimension is an invariant under any isomorphism of algebraic groups. It is possible, however, to have the same abstract group arising as algebraic groups of different dimensions over a given field.
- Given an algebraic group over a field extension of , naturally acquires the structure of an algebraic group over . The dimension of over is the product of the dimension of over and the degree of the extension .
`Further information: formula for dimension for change of base field of algebraic group` - The dimension of an algebraic group equals the dimension of its connected component of identity.

## Particular cases

- The trivial group, viewed as an algebraic group over any field, is zero-dimensional.
- The additive group of a field, as well as the multiplicative group of a field, are both one-dimensional as algebraic groups under the usual structure.
- The additive group of a -dimensional vector space has dimension as an algebraic group.
- The general linear group over a field of degree has dimension as an algebraic group.

## Effect of operations

Operation | Input groups and their orders | Output group and its order | Proof and comment |
---|---|---|---|

external direct product of two algebraic groups | has dimension , has dimension | has dimension | dimension of direct product is sum of dimensions; the same formula works for internal direct product, which is equivalent to external direct product. |

external direct product of finitely many algebraic groups | with orders respectively | has order | dimension of direct product is sum of dimensions; same formula works for internal direct product |

external semidirect product of two groups | , dimension , , dimension , acting on it via algebraic automorphisms | has dimension | dimension of semidirect product is sum of dimensions; same formula works for internal semidirect product |

group extension | closed normal subgroup , dimension , quotient group , dimension | dimension of extension is sum of dimensions of normal subgroup and quotient |