One-parameter subgroup

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

It is a special kind of curve through the identity.

There is a bijection between one-parameter subgroups and the tangent space at identity (the Lie algebra of $G$). This bijection is given by the exponential map.

Outline:

• In one direction, given a one-parameter subgroup $\varphi$, we take $\varphi ' (0)$ .

• Conversely, given $V\in \mathfrak{g}$, we consider the left-invariant vector field $V^\#$ in $G$. This relation is 1-1.

• This vector field gives rise to a flow $\varphi(t,m)$ which is a one-parameter local group of transformations. Since left-invariant vector fields are complete (see proof here, we can take $g_t=\phi(t,e)$.

• It can be shown that $\varphi(t,m)=m\cdot g_t$ (see here) and this assignment $t\mapsto g_t$ is the corresponding one-parameter subgroup.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es