# Pushforward of differential form on Lie Group

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.

• Read about pullback of differential forms. Commented Jun 18, 2020 at 1:16

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 $$w(Y_1,\dots,Y_k)$$.