Lie groups ed

Definition ed

Lie group \[G\]

A smooth manifold, with smooth multiplication \[\mu : G \times G \rightarrow G\] and inversion \[i : G \rightarrow G\].

Left translation

Diffeomorphism \[L_g : G \rightarrow G, L_g(a) = ga \]

Lie algebra \[\mathfrak{g}\]

Tangential space at the identity: \[\mathfrak{g} = T_e G\]...

Maurer-Cartan form ed

Definition ed

Maurer-Cartan form \[\omega\]

Left-invariant \[\mathfrak{g}\]-valued 1-form \[\omega : TG \rightarrow \mathfrak{g}\] with \[ \omega(v) = L_{g^{-1}\star} (v) \] for \[v \in T_gG\].

"moving a tangential vector from point \[g\] to \[e\] via the natural isomorphism of \[L_{g^{-1}}\]."

For matrices \[M \in GL(n)\], the Maurer-Cartan form is \[\omega = M^{-1} dM\].

Algegra-valued forms ed

For a basis \[E^i\] of the Lie algebra \[\mathfrak g\], we can define \[\mathfrak g\]-valued forms as a product of regular forms

\[ \omega_i E^i \]

This hints towards a new product

\[ [\mu \wedge \nu] := \mu_i \wedge \nu_k [E^i, E^k] \]

For \[\mathfrak g\]-valued 1-forms, we have

\[ 2 [\omega(X), \omega(Y)] = \omega_i(X) \omega_k(Y) [E^i, E^k] + \omega_i(X) \omega_k(Y) \underbrace{[E^i, E^k]}_{-[E^k, E^i]} \\ \qquad = (\omega_i(X) \omega_k(Y) - \omega_k(X) \omega_i(Y)) [E^i, E^k] \\ \qquad = (\omega_i \wedge \omega_k)(X, Y) [E^i, E^k] \\ \qquad = [\omega \wedge \omega] (X,Y) \]

Structure equations ed

\[ d\omega + \frac{1}{2} [\omega \wedge \omega] = 0 \]

proof

in general \[ d\omega(X,Y) = X \omega(Y) + Y \omega(X) - \omega([X,Y]) \]

taking \[X,Y\] left-invariant, \[\omega(Y) = \operatorname{const}\] and so \[X \omega(Y) = Y \omega(X) = 0\]

\[[X,Y]\] is also left-invariant, so \[\omega([X,Y]) = [X,Y]_e = [X_e, Y_e] = [\omega(X), \omega(Y)] = \frac{1}{2} [\omega \wedge \omega](X,Y)\]

since the form \[ d\omega + \frac{1}{2} [\omega \wedge \omega] \] vanishes for all left-invariant vector fields, it has to vanish in general