Ivancevic_Applied-Diff-Geom

204 Applied Differential Geometry: A Modern IntroductionFor every g in a Lie group G, the two maps,L g : G → G, h ↦→ gh, andR h : G → G,g ↦→ gh,are called left and right translation maps. Since L g ◦ L h = L gh , and R g ◦R h = R gh , it follows that (L g ) −1 = L g −1 and (R g ) −1 = R g −1, so both L gand R g are diffeomorphisms. Moreover L g ◦ R h = R h ◦ L g , i.e., left andright translation commute.A vector–field X on G is called left–invariant vector–field if for everyg ∈ G, L ∗ gX = X, that is, if (T h L g )X(h) = X(gh) for all h ∈ G, i.e., thefollowing diagram commutes:T G✻XT L g✲ T G✻XGL g✲ GThe correspondences G → T G and L g → T L g obviously define a functorF : LG ⇒ LG from the category G of Lie groups to itself. F is a specialcase of the vector bundle functor (see (4.3.2) below).Let X L (G) denote the set of left–invariant vector–fields on G; it is aLie subalgebra of X (G), the set of all vector–fields on G, since L ∗ g[X, Y ] =[L ∗ gX, L ∗ gY ] = [X, Y ], so the Lie bracket [X, Y ] ∈ X L (G).Let e be the identity element of G. Then for each ξ on the tangent spaceT e G we define a vector–field X ξ on G byX ξ (g) = T e L g (ξ).X L (G) and T e G are isomorphic as vector spaces. Define the Lie bracket onT e G by[ξ, η] = [X ξ , X η ] (e),for all ξ, η ∈ T e G. This makes T e G into a Lie algebra. Also, by construction,we have[X ξ , X η ] = X [ξ,η] ,this defines a bracket in T e G via left extension. The vector space T e G withthe above algebra structure is called the Lie algebra of the Lie group G andis denoted g.