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