One-parameter subgroup

A smooth homomorphism from (R,+) into a Lie group, generated by a Lie algebra element.

One-parameter subgroup

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ .

Definition

A one-parameter subgroup of $G$ is a smooth group homomorphism

\gamma:(\Bbb R,+)\to G.

Equivalently, $\gamma$ is a smooth curve satisfying $\gamma(t+s)=\gamma(t)\gamma(s)$ and $\gamma(0)=e$ .

For each $X\in\mathfrak g$ , the curve

\gamma_X(t)=\exp(tX)

Conversely, if $\gamma$ is any one-parameter subgroup, then the derivative

X=\gamma'(0)\in T_eG\cong \mathfrak g

determines $\gamma$ uniquely by $\gamma(t)=\exp(tX)$ for all $t$ .

Thus, one-parameter subgroups are in bijection with elements of $\mathfrak g$ .

One-parameter subgroups are the group-theoretic shadows of constant-coefficient ODE flows on $G$ ; the precise relationship is expressed by their interpretation as integral curves of invariant vector fields .