One-parameter subgroup

In the theory of topological groups , a one-parameter subgroup is a continuous group homomorphism from the additive group of real numbers into a topological group. The image of a one-parameter subgroup is a subgroup in the group-theoretical sense.

One-parameter subsets of Lie groups

If a Lie group , then a map is a one-parameter subgroup if the map is smooth and a group homomorphism. For homomorphisms between Lie groups, smoothness is equivalent to continuity. Each one-parameter subgroup corresponds to exactly one element in the Lie algebra of . Depending on the approach, the Lie algebra is sometimes even defined as the set of one-parameter subgroups. ${\displaystyle G}$ ${\displaystyle \varphi :\mathbb {R} \rightarrow G}$ ${\displaystyle G}$

Examples

• The continuous group homomorphisms of the additive group of real numbers in themselves are exactly the mappings for a solid one . ${\displaystyle \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x\mapsto \lambda x}$ ${\displaystyle \lambda \in \mathbb {R} }$
• The continuous group homomorphisms from the additive group of real numbers into the multiplicative group of non-zero real numbers are exactly the mappings for a fixed one . ${\displaystyle \mathbb {R} \to \mathbb {R} ^{\times }}$ ${\displaystyle x\mapsto a^{x}}$ ${\displaystyle a\in \mathbb {R} _{>0}}$

literature

• John Frank Adams , Lectures on Lie groups , Benjamin, 1969