# 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]]