Topics in Geometric Mechanics: Week 4

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Week 4

This week will look at

• induced linear transformations
• Lie groups
• orbits of Lie groups

Induced Linear Transformations

Given $A : V \to W$ linear, there is an induced transformation \[ A : \tenalg{*}{V} \to \tenalg{*}{W} \] that preserves the grading and symmetry of a tensor. Given $v \in V$ and $\eta \in \tenalg{k}{V}$, we define \begin{align*} \tenalg{k+1}{A} &: \tenalg{k+1}{V} \to \tenalg{k+1}{W} \\ v \tensorproduct \eta &\mapsto A(v) \tensorproduct \tenalg{k}{A}(\eta) \\ \tenalg{1}{A} &= A. \end{align*}

Facts

Here are several facts about the induced transformation

• $A$ is invertible iff $\tenalg{*}{A}$ is.
• $A$ is diagonalizable iff $\tenalg{*}{A}$ is.
• If $A$ has eigen-pairs $(e_i,\lambda_i)$, then $\tenalg{k}{A}$ has $(e_I,\lambda_I)$ eigen-pairs for all $k$-multisets of $\set{1, \cdots, n}$.

Lie groups

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 \[ \iota(g) = g^{-1}. \]

Examples of Lie Groups

Here is a non-exhaustive list:

• every finite group
• every discrete group
• $\R^n$, $\C^n$
• $\GL{n;\R}$, $\GL{n;\C}$
• $\Orth{n;\R}$, $\Orth{n;\C}$
• $\SL{n;\R}$, $\SL{n;\C}$
• every closed subgroup of a Lie group is a Lie group (!)
• If $G$ is a Lie group and $H \lhd G$ is a closed, normal subgroup, then $G/H$ is a Lie group.
• a finite Cartesian product of Lie groups is a Lie group

The Torus

Consider the groups:

\begin{align*} \Unitary{1} &= \set{ z \in \C \st \norm{z}=1 } \\ \SOrth{2} &= \set{ \begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix} \st \theta \in \R } \\ \sphere{1} &= \R/\Z. \end{align*}

These Lie groups are Lie group isomorphic.

The Torus — continued

Consider the groups:

\begin{align*} \Diagonal{n} &= \set{ \diag{z_1, \ldots, z_n} \st \norm{z_1}=\cdots=\norm{z_n}=1} \\ \Delta &= \set{ \diag{R_{\theta_1}, \ldots, R_{\theta_n}} \st R_\theta = \begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix}, \theta \in \R } \\ \Torus{n} &= \R^n/\Z^n. \end{align*}

These Lie groups are Lie group isomorphic.