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