By Werner Hildbert Greub
Greub W., Halperin S., Vanstone R. Connections, curvature, and cohomology. Vol.2.. Lie teams, valuable bundles, and attribute sessions (AP, )(ISBN 0123027020)(543s)_MD_
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Extra info for Connections, curvature and cohomology. Volume 2, Lie groups, principal bundles and characteristic classes
Definition: Let h E Te(G). T h e unique left invariant vector field X such that X ( e ) = h is denoted by X,,and is called the left invariant vector field generated by h. Similarly, a vector field Y is called right invariant if (pb)*Y= Y , b E G. T h e Lie algebra of right invariant vector fields is denoted by T R ( G ) The . same proof as given in Proposition I shows that Y w Y(e) defines an isomorphism XR(G) 5 Te(G). T h e right invariant vector field corresponding to h E Te(G)under this isomorphism is called the right invariant vectorfieldgenerated by h, and is denoted by Yh.
In view of sec. 2, we have [h, k] = h, k -[h, k]", E Te(G). Thus the map h ++ -h defines an. ) H . Since Now consider a homomorphism of Lie groups, s,: G y(e) = e ( e denotes the unit of both groups), the derivative ds, restricts to a linear map --f (dq)e: Te(G) + This map will be denoted by s,'. Te(H). I. Lie Groups 28 Proposition 111: v' is a homomorphism of Lie algebras. Proof: I t follows from sec. 1 that Hence [ X h, X,] 7 [X,,, , Xvtk]. Evaluate this relation at e to obtain v"h, k] = [v'h, v'kl.
T h e cross-sections in the bundle L(AT$ ; M x F ) (respectively, L(APT$ ; M x F)) are called differential forms with values in F (respectively, p-forms with values in F); these modules are denoted by A ( M ;F ) and AP(M;F). If Q E AP(M;F), then Q(x) is a skew-symmetric, p-linear, F-valued function in T,(M). a, 69) . 0. T h e operators i(X) @ L, O(X) @ L, and 6 @ 6 in A ( M ;F ) are denoted simply by i(X), O(X),and 6; they satisfy the relations given above in the case F = R. A smooth map v: M --+ N induces a map lp* = (v* 0L): A ( M ;F ) + A ( N ;F ) .