Topics in Geometric Mechanics: Week 5

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Week 5

Last week we looked at 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

Translations and Conjugations

For each $g \in G$, the maps \begin{align*} L_g x &= g \cdot x && L_g : G \to G \\ R_g x &= x \cdot g && R_g : G \to G \\ I_g x &= g \cdot x \cdot g^{-1} && I_g : G \to G. \end{align*} are diffeomorphisms of $G$.

1-parameter subgroups

Let $\phi : \R \to G$ be a Lie group homomorphism:

• $\phi$ is a smooth map;
• $\phi(s+t) = \phi(s) \phi(t)$ for all $s, t \in \R$.

Examples

Here are examples of 1-parameter subgroups:

• $\phi : \R \to \R$ given by $\phi(t) = t$.
• $\phi : \R \to \Torus{1}$ given by $\phi(t) = t \bmod \Z$.
• $\phi : \R \to \R^n$ given by $\phi(t) = t v$.
• $\phi : \R \to \Torus{n}$ given by $\phi(t) = t v \bmod \Z^n$.

More Examples

Here are examples of 1-parameter subgroups:

• $\phi : \R \to \GL{\R^n}$ given by $\phi(t) = \exp(t A)$.
• $\phi : \R \to \C^*$ given by $\phi(t) = e^{t z}$.

1-parameter subgroups

Let $\phi : \R \to G$ be a Lie group homomorphism:

• There is a bijection between 1-parameter subgroups and $T_1G$.
• Let $\exp : T_1G \to G$ be the map $\xi \mapsto \phi_{\xi}(1)$.
• Here is a key fact:

  The exponential map has $\Dat{0}{\exp} = 1_{T_1G}$. In addition, there is an open neighbourhood of $U \ni 0 \in T_1G$ such that $\exp|U$ is a diffeomorphism.

Adjoint action

Let $g \in G$. Define, for $x \in T_1G$, \begin{align*} \Ad{g}{x} &= \left.\ddt{t}{}\right|_{t=0} g \exp(tx) g^{-1}, && \Ad{g} : T_1G \to T_1G \end{align*} $\Ad{g} = \Dat{1}{I_g}$.

adjoint action

For $x \in T_1G$, define \begin{align*} \ad{x}{y} &= \left.\ddt{t}{}\right|_{t=0} \Ad{\exp(tx)}{y} && \ad{x}{} : T_1G \to T_1G. \end{align*}

Left- and Right-Invariant Vector Fields

• We define a vector field $X$ on $G$ to be left-invariant if $X(g) = \Dat{1}{L_g} X(1)$ for all $g \in G$.
• We define a vector field $X$ on $G$ to be right-invariant if $X(g) = \Dat{1}{R_g} X(1)$ for all $g \in G$.
• An invariant vector field is determined by its value at $1$.
• Let $\liealg{g}_{\pm}$ be the set of right/left invariant vector fields on $G$.

Lie Bracket

• Define the Lie bracket of vector fields $X, Y$ to be in local coordinates $q^i$ \begin{align*} \liebracket{X}{Y} &= X \circ Y - Y \circ X = \sum_{i, j} X^j \didi{Y^i}{q^j} \didi{\ }{q^i} - Y^j \didi{X^i}{q^j} \didi{\ }{q^i} \end{align*}
• Example: $X(q) = Aq$ and $Y(q) = Bq$ for $q \in \R^n$. Then \begin{align*} \liebracket{X}{Y}(q) &= -(AB-BA)q \end{align*}
• Example: $G = \GL{R^n}$ and $T_gG = \Mat{n}{\R}$. The vector fields $X(g) = gA$ and $Y(g) = gB$ are left-invariant. \begin{align*} \liebracket{X}{Y}(g) &= g(AB-BA). \end{align*}