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I'm extremely confused by what's going on in section 10.1.3, pg 377 of the 2nd edition of Nakahara, in regards to his notation for lie algebra-valued one forms.

We let [tex] \{U_i\} [/tex] be an open covering of a smooth manifold [tex] M [/tex] and let [tex] \sigma_i [/tex] be a local section of M into the principle G-bundle P defined on each element in the covering. We also let [tex]A_i[/tex] be a Lie-algebra-valued one form on each [tex]U_i.[/tex]

We then want to construct a Lie-algebra-valued one-form on P using this data, and we do so by defining [tex] \omega_i \equiv g_i^{-1}\pi^*A_ig_i + g_i^{-1}\mathrm{d}_Pg_i[/tex] where [tex]d_P[/tex] is the exterior derivative on P and [tex]g_i[/tex] is the canonical local trivialization defined by [tex]\phi_i^{-1}(u)=(p,g_i)[/tex] for [tex]u=\sigma_i(p)g_i.[/tex]

What I don't understand is the second term in the definition of [tex]\omega_i.[/tex] I don't understand what the exterior derivative is acting on there. He writes it as if it's acting directly on the group element [tex]g_i\in G[/tex] and later writes terms like [tex]\mathrm{d}_Pg_i(\sigma_{i*}X),[/tex] where [tex]X\in T_pM[/tex] and [tex]\sigma_{i*}[/tex] is the push-forward so that [tex] \sigma_{i*}X \in T_{\sigma_i(p)}P[/tex] as if [tex]\mathrm{d}_Pg_i[/tex] is itself a Lie algebra-valued one-form, which I simply don't understand at all. It just doesn't seem well defined and I see no way of making sense of that expression. Thanks so much in advance for any clarification as to what that term is doing, i.e., how it takes in a vector in the tangent space of some point in P and spits out an element of the lie algebra of G.

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# Notation in Ch. 10 of Nakahara

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