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

Maurer-Cartan form \[\omega_G\]

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

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