Dimension of an algebraic group

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 $G$ over a field extension $L$ of $K$, $G$ naturally acquires the structure of an algebraic group over $K$. The dimension of $G$ over $K$ is the product of the dimension of $G$ over $L$ and the degree of the extension $L/K$. 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 $n$-dimensional vector space has dimension $n$ as an algebraic group.
• The general linear group over a field of degree $n$ has dimension $n^{2}$ 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	$G$ has dimension $m$, $H$ has dimension $n$	$G\times H$ has dimension $m+n$	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	$G_{1},G_{2},\dots ,G_{n}$ with orders $m_{1},m_{2},\dots ,m_{n}$ respectively	$G_{1}\times G_{2}\times \dots \times G_{n}$ has order $\sum _{i=1}^{n}m_{i}=m_{1}+m+2+\dots +m_{n}$	dimension of direct product is sum of dimensions; same formula works for internal direct product
external semidirect product of two groups	$G$, dimension $m$, $H$, dimension $n$, acting on it via algebraic automorphisms	$G\rtimes H$ has dimension $m+n$	dimension of semidirect product is sum of dimensions; same formula works for internal semidirect product
group extension	closed normal subgroup $N$, dimension $m$, quotient group $G/N$, dimension $n$	$m+n$	dimension of extension is sum of dimensions of normal subgroup and quotient