Invariant Forms on a Lie Group


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w is left-invariant if it's pullback under left multiplication by g is itself, I.e the pullback of the k-covector w_gx is w_x. Thus w_g = pullback of w_e under left multiplication by g^{-1}, is completely determined by w_e Every left invariant k-form is smooth. To check that, it suffices to show that w(X1,..,Xn) is smooth for smooth vector fields X_i. If Y_i are left invariant vector fields generated by a basis of TeG then X_j are a linear combination of them over smooth functions so suffices to check w(Y...) is smooth. The latter is constant and hence smooth. The space of left-invariant k-forms on G is isomorphic to the space of k-covectors on TeG and hence has dim n chooses k. Note: TpN has dim n since locally at p, N looks like Rn.

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