Topics in Geometric Mechanics: Week 6

Week 6

Last week we looked at Lie groups:

A Lie group is a group $G$ that satisfies

$G$ is a smooth manifold
Multiplication $\mu : G \times G \to G$ is a smooth map \[ \mu(g_1, g_2) = g_1 \cdot g_2. \]
Inversion $\iota : G \to G$ is a smooth map \[ \iota(g) = g^{-1}. \]

Lie Group Actions

An action of a Lie group $G$ on a manifold $M$ is a smooth map $\alpha : G \times M \to M$ s.t. \[ \alpha(h, \alpha(g, x)) = \alpha(hg, x) \] for all $h, g \in G$, $x \in M$. One often writes $g \cdot x$ for $\alpha(g, x)$.

Lie Group Actions - Examples

$G$ acting on $M = G$ by left translations
$G$ acting on $M = G$ by $g \cdot x = xg^{-1}$
$G = H \times H$ acting on $M = H$ by $(h_0, h_1) \cdot x = h_0 x h_1^{-1}$
$G = \GL{\R^n}$ acting on $M=\R^n$ by its standard linear representation
$G = \GL{V}$ acting on $\tenalg{k}{V}$ by the induced transformations
$G = \Orth{\R^n}$ acting on $M = \sphere{n-1} \subset \R^n$ by linear transformations
$G$ acting on $\liealg{g}$ (its Lie algebra) by the adjoint action

The action on the sphere

Let us see that $G = \Orth{\R^n}$ acting on $\sphere{n-1} = \set{ x \in \R^n \st \norm{x}=1 }$ is a Lie group action.

The map $m : \Mat{n}{\R} \times \R^n \to \R^n$ given by \[ m(A, x) = Ax \] is smooth and satisfies $m(A, m(B, x)) = m(BA, x)$.
The inclusion map $\iota : \GL{n;\R} \hookrightarrow \Mat{n}{\R}$ is smooth.
The composition $\mu = m \circ (\iota \times id) : \GL{n;\R} \times \R^n \to \R^n$ is smooth and satisfies $\mu(h, \mu(g, x)) = \mu(hg, x)$.
The inclusion $\zeta : \Orth{\R^n} \hookrightarrow \GL{n;\R}$ is a Lie group homomorphism.
The map $\eta = \mu \circ (\zeta \times id) : \Orth{\R^n} \times \R^n \to \R^n$ is a Lie group action.

Facts about Actions

Let $\alpha : G \times M \to M$ be a Lie group action.

If $H \lt G$ is a closed subgroup, then the restriction $\alpha_H : H \times M \to M$ is a Lie group action.

If $N \subset M$ is a smooth submanifold invariant under $G$ (i.e. $g \cdot n \in N$ for all $n \in N, g \in G$), then the restriction $\alpha_N : G \times N \to N$ is a Lie group action.

Let $m \in M$. Let $\stab{G}{m} = \set{ g \in G \st g \cdot m = m }$ be the stabiliser subgroup.

Let $\orbit{G}{m} = \set{ g \cdot m \st g \in G }$ be the orbit of the point $m$.

For each $x = g \cdot m \in \orbit{G}{m}$, the stabiliser $\stab{G}{x} = g \stab{G}{m} g^{-1}$.
If $G$ is compact, then $\orbit{G}{m}$ is a smooth submanifold with a transitive Lie group action of $G$. It is diffeomorphic to $G/H$ where $H = \stab{G}{m}$.

For each subgroup $H \leq G$, let \[ M_H = \set{ m \in M \st \stab{G}{m} \text{ is conjugate to } H }. \]
$M$ is a disjoint union \[ M = \coprod_{H \leq G} M_H. \]
If $G$ is compact, then $M_H$ is a smooth submanifold with a smooth Lie group action of $G$ on it for all $H$.

An Example

Let $M = \sphere{2} \subset \R^3$ and let \[ G = \Orth{\R^1} \times \Orth{\R^2}. \]

$\stab{G}{N} = \stab{G}{-N} = 1 \times \Orth{\R^2}.$
$\stab{G}{E} = \Orth{\R^1} \times \Orth{\R^1} \times 1.$
$\stab{G}{X} = 1 \times \Orth{\R^1} \times 1.$

The orbits and stabilizers of points on the 2-sphere.

A Second Example

Let $G = \GL{\C^3}$ act on $M = \Mat{3}{\C}$ by \[ g \cdot x = gxg^{-1}. \]

By the Jordan canonical form theorem, every $x \in \Mat{3}{\R}$ is conjugate to one of ($a, b, c \in \C^{\times}$ are distinct)
- Semisimples:
  
  \begin{align*} D &= \begin{bmatrix} a && \\ &b& \\ &&c \end{bmatrix} && E = \begin{bmatrix} a && \\ &a& \\ &&c \end{bmatrix} && F = a I \end{align*}
- Non-semisimples:
  \begin{align*} U &= \begin{bmatrix} a & 1 & \\ &a& \\ &&b \end{bmatrix} && V = \begin{bmatrix} a & 1 & \\ & a & 1 \\ &&a \end{bmatrix} \end{align*}

The stabilisers of each element type:

$\stab{G}{D} = A = \set{ \text{all diagonal elements} }.$ (``complex torus'')
$\stab{G}{E} = \GL{\C^2} \times \GL{\C^1} = \set{ x = \begin{bmatrix} g & \\ & c \end{bmatrix} \st g \in \GL{\C^2}, c \in \C^{\times} = \GL{\C^1} }$.
$\stab{G}{F} = G$.
$\stab{G}{U} = N_1 \times \GL{\C^1} = \set{ x = \begin{bmatrix} s&t& \\ &s& \\ &&c \end{bmatrix} \st s, c \neq 0 }$.
$\stab{G}{U} = N_2 = \set{ x = \begin{bmatrix} s&t& \\ &s&t \\ &&s \end{bmatrix} \st s \neq 0 }$.

A Third Example

Let $M = \Torus{2} = \R^2/\Z^2$ be the $2$-dimensional torus. Let $G = \R v/\Z^2$ be the $1$-parameter subgroup of $\Torus{2}$ generated by $v = (1, \sqrt{2})$.

$G$ acts freely on $M$.
$\orbit{G}{x}$ is dense in $M$; it is never an embedded submanifold.

Intervals of increasing length in the 1-parameter subgroup.