Dimension of an algebraic group
|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.|
- 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.
- 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|