# Definition
Given a [[Lie group]] $G$ and its [[Lie Algebra]] $\mathfrak{g}$. Let $X \in \mathfrak{g}$ and define a [[group Homomorphism]] from the real additive group $\mathbb{R}$ to the [[Matrix Lie Group]] $G$ as follows:
$
\begin{align}
\exp: \mathbb{R} &\rightarrow G\\
t & \mapsto e^{tX}
\end{align}
$
This homomorphism defines the one-parameter subgroup.
Conversely, consider a one-parameter subgroup $\gamma: \mathbb{R} \rightarrow G$, then by the properties of homomorphisms, we have $\gamma(0) = I$. Defining the element of the [[Lie Algebra]]:
$
X \equiv \frac{d \gamma}{dt}(0)
$
Then
$
\begin{align}
\frac{d\gamma}{dt}(t) &= \lim_{h\rightarrow0} \frac{\gamma(h+t)-\gamma(t)}{h},\\
&= \lim_{h\rightarrow0} \frac{\gamma(h)\gamma(t)-\gamma(t)}{h},\\
&= \lim_{h\rightarrow0} \frac{\gamma(h)-I}{h}\gamma(t),\\
&= \lim_{h\rightarrow0} \frac{\gamma(h)-\gamma(0)}{h}\gamma(t),\\
&= X\gamma(t),\\
\end{align}
$
therefore
$
\gamma(t) = e^{tX}
$
and we have proved that every one parameter subgroup is of the form:
$
\begin{align}
\exp: \mathbb{R} &\rightarrow G\\
t & \mapsto e^{tX}
\end{align}
$
# Example: $SO(3)$
Consider the matrix for a [[Rotation]] about the $z$ axis, an [[group element|element]] of [[Special Orthogonal Group in 3 Dimensions|SO(3)]]:
$
R_{z}(\theta)=\left(\begin{array}{ccc}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right)
$
Now let the rotation angle be $\epsilon \ll 1$, we can write:
$
R_z(\epsilon) \approx R_z(0) + \epsilon \frac{dR_z}{d\theta}\bigg|_{\theta = 0} = I + \epsilon L_z
$
where $L_z$ is an element of the [[Lie Algebra]] of $SO(3)$, denoted $\mathfrak{so}(3)$:
$
L_z \equiv \frac{dR_z}{d\theta}\bigg|_{\theta = 0} =
\begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
$
Given a finite rotation by an angle $\theta$, we should be able to describe it as a series of $n$ rotations by an angle $\theta/n$. Assuming $n$ is large, we set $\epsilon = \theta/n$ in the expression above and we get:
$
R_z(\theta) \approx \left(I + \frac{\theta L_z}{n}\right)^n
$
this becomes an equality in the limit $n \rightarrow \infty$
$
R_z(\theta) = \lim_{n\rightarrow \infty} \left(I + \frac{\theta L_z}{n}\right)^n = e^{\theta L_z}
$
The set $\{R_z(\theta) = e^{\theta L_z}|\theta \in \mathbb{R}\}$ is a [[subgroup]] of $SO(3)$. If we rewrite this as:
$
R_z : \mathbb{R} \rightarrow SO(3)
$
then $R_z$ is a [[group homomorphism|homomorphism]] from the additive group $\mathbb{R}$ to the [[Matrix Lie Group]] $SO(3)$, which is a one-parameter subgroup as defined above.
In fact, the set of rotations about any particular axis also forms a one-parameter subgroup.
# Example 2: $SO(3,1)_o$
In the [[Restricted Lorentz Group in 4 Dimensions]], there is a one-parameter subgroup consisting of all boosts along an arbitrary direction.
# In Terms of Manifolds
We can think of the one-parameter subgroups $\gamma(t) = e^{tX}$ as parameterizing a set of [[Curve|curves]] in the group [[manifold]], which tells us that $X$ is the [[tangent vector]] to $\gamma(t)$ at the identity (since $X \equiv \frac{d \gamma}{dt}(0)$). If we interpret all such $X$ this way, then we see that they compose the entire [[tangent space]] to $G$ at $e$, as shown schematically in the figure.
![[Pasted image 20210704173548.png|400]]