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Let G be a Lie Group and let L be the tangent space at identity. Further let w be a form on L,that is,a wedge of contangent vectors at the identity.

Is there a way to define pushforward of w via left translation maps $L_g$ over G?

My problem is that taking derivative takes me to tangent space but here I want to go to (alternating product of) cotangent space.

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    $\begingroup$ Read about pullback of differential forms. $\endgroup$ Commented Jun 18, 2020 at 1:16

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This is more like a pullback, you just transport tangent vectors back to the origin. It is easier to view $w$ as a multilinear map than as a wedge of cotangent vectors (which is not the most general possibility). Denoting by $\lambda_g$ the left translation by $g$, you define $$ \tilde w(g)(X_1,\dots,X_k):=w(T_g\lambda_{g^{-1}}(X_1),\dots,T_g\lambda_{g^{-1}}(X_k)) $$ for $X_1,\dots,X_k\in T_gG$, which gives $\tilde w\in\Omega^k(G)$. Alternatively, $\tilde w$ is characterized by the fact that for left invariant vector fields $L_{Y_i}$ the function $\tilde w(L_{Y_1},\dots,L_{Y_k})$ is constant and equal to $w(Y_1,\dots,Y_k)$.

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