Lie Groups and Lie Algebras Lecture 5 notes

Definition: Preliminaries

Consider a smooth manifold $M$ with a smooth vector field $\xi$. Then, the vector field defines a system of ODEs on M: $\frac{d}{dt}x = \xi(t)$
A smooth curve $\gamma(t), t \in (-\epsilon, \epsilon)$ is an integral curve of $\xi$ if $\frac{d}{dt}\gamma(t) = \xi(\gamma(t))$ (in other words, $\gamma(t)$ is a solution of $\frac{d}{dt}x = \xi(t)$)
Existence and uniqueness theorem: For any $x \in M$ there is a unique integral curve $\gamma_x(t)$ passing through it (s.t. $\gamma_x(0) = x$).
Flow $\phi_t$: To each vector field $\xi$, we can assign (at least locally) a diffeomorphism $\phi_t$ which shifts each point $x$ along $\xi$ by time $t$. In other words, $\phi_t(x) = \gamma_x(t)$.
"Locally" means that usually the flow $\phi_t$ is defined only for sufficiently small $t$ (depends on $x$).
Completeness: If each integral curve $\gamma_x(t)$ can be extended (in the sense of $t$) to $\mathbb{R} = (-\infty, \infty)$, then $\xi$ is called complete. Equivalently, completeness of $\xi$ means that the flow $\phi_t: M \to M$ is globally defined for all $t \in \mathbb{R}$.
Since $\phi_t\circ phi_s = \phi_{t + s}$, the flow can be thought of as a group under composition.
The differential of a smooth map $F: M \to N$ is the map $dF: TM \to TN$ defined by $dF(\frac{d}{dt}\gamma(t)) = \frac{d}{dt}F(\gamma(t))$. (sends tangent vectors to tangent vectors)
Lie bracket of vector fields: Given smooth vector fields $\xi$ and $\eta$ on $M$, we can introduce a new vector field $[\xi, \eta]$ s.t.:
- $[\xi, \eta]^k = \xi^i \frac{d}{d x^i} \eta^k - \eta^i \frac{d}{d x^i} \xi^k$ (in local coords)
- $[\xi, \eta](f) = \xi(\eta(f)) - \eta(\xi(f))$
.#-> This lie bracket represents taking the differential of $\xi$ with respect to $\eta$, or is it the other way around?
If $F: M \to N$ is a diffeomorphism, with $dF: TM \to TN$ its differential, then $dF([\xi, \eta]) = [dF(\xi), dF(\eta)]$ #-> This is obvious because $F$ is a transformation of local coords and the Lie bracket is independent on local coords?
Vector fields $\xi, \eta$ commute (i.e. $[\xi, \eta] = 0$) if and only if the corresponding flows commute (i.e. $\phi_{\eta}\circ \phi_{\xi} = \phi_{\xi}\circ \phi_{\eta}$)

Definition: Left Invariance

A vector field $\xi$ is called left invariant if it is preserved under left translations.
In other words, for any $a \in G$, $dL_a(\xi(x)) = \xi(L_a(x))$.
Similarly, a vector field $\eta$ is called right invariant if $dR_a(\eta) = \eta$ for any $a \in G$.
In other words, if we consider the values $\xi(x)$ and $\xi(y)$ of our vector field $\xi$ at two distinct points $x \in G$ and $y = ax \in G$, then $dL_a(\xi(x)) = \xi(ax)$. Similar for right invariance.

Properties:
If $\xi$ is a left invariant vector field on $G$, then
Proposition 1:
Let $\gamma_e(t)$ be the integral curve of $\xi$ passing through the identity $e$ (i.e. $\gamma_e(0) = e$). Then, the integral curve $\gamma_x(t)$ of $\xi$ passing through $x$ is $x \gamma_e(t) = L_x(\gamma_e(t))$.
Proof:
$\frac{d}{dt} L_x(\gamma_e(t)) = dL_x(\frac{d}{dt} \gamma_e(t))$ by def of differential, $dL_x(\frac{d}{dt} \gamma_e(t)) = dL_x(\xi(\gamma_e(t))$ by $\gamma_e(t)$ being an integral curve of $\xi$, and = $\xi(L_x(\gamma_e(t))$ by $\xi$ being left-invariant.
Thus, $L_x(\gamma_e(t)) = x \gamma_e(t)$ is an integral curve of $\xi$. It suffices that $x \gamma_e(0) = xe = x$ to show that $x \gamma_e(t)$ passes through $x$.
Corollary:
The left translation of any integral curve of $xi$ is again an integral curve
Corollary:
The flow $\phi_{\xi}^t: G \to G$ of $xi$ is defined by $\phi_{\xi}^t(x) = x \gamma_e(t)$, where $\gamma_e(t)$ is the integral curve of $\xi$ through the identity.
Proposition 2:
$\xi$ is complete, i.e. the flow $\phi_{\xi}^t: G \to G$ of $\xi$ is well defined for all $t \in \mathbb{R}$
Proof:
The definition of the flow for a left-invariant vector field $\xi$ is defined as $\phi_{\xi}^t(x) = x \gamma_e(t)$. If $\gamma_e(t)$ is defined on $(-\epsilon, \epsilon)$, then $\phi_{\xi}^t$ is defined on the whole group for $t \in (-\epsilon, \epsilon)$.
Then, $\phi^t$ can naturally be defined for all $t \in (-\infty, \infty)$ just by iterating: $\phi_{\xi}^t = \phi_{\xi}^{\frac{t}{k}}\circ\dots\circ\phi_{\xi}^{\frac{t}{k}}$ $k$ times where $\frac{t}{k} \in (-\epsilon, \epsilon)$.

Definition: One-parameter Subgroup

Definition: A smooth map $f: \mathbb{R} \to G$ is called a one-parameter subgroup of $G$ if $f(s + t) = f(s)f(t)$ for any $t, s \in \mathbb{R}$ #-> i.e. it can be parameterized by $\mathbb{R}$