Dimension of an algebraic group

From Groupprops
Jump to: navigation, search

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

Particular cases

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