Maurer-Cartan Form: Is it a One Form?

[Silviu]

Hello! I am reading Geometry Topology and Physics by Nakahara and in Chapter 5.6.4 he defines the canonical (Maurer-Cartan) one form on a Lie group G as: $\theta : T_gG \to T_eG$. Then he states a theorem in which $\theta = V_\mu \otimes \theta^\mu$. Both by the tensor product and by the definition from a vector space to another vector space, $\theta$ seems to be a $(1,1)$ tensor, not a one form, as it is stated in the book. Am I missing something? Is it a one form or not? Did I get wrong the definition of one form? Thank you!

 

[martinbn]

It is a $\mathfrak g$ valued one form.

 

[fresh_42]

Every $(1,1)$ tensor, say $\varphi \otimes v \in U^* \otimes V$ is equivalent to the linear map $u \longmapsto \varphi(u)\cdot v$. Here we have $v \in T_gG$ and $\varphi \in T_e^*G$.

 

[Silviu]

It is a $\mathfrak g$ valued one form.

Isn't a one-form by definition a function from vectors to real numbers? And $\mathfrak{g}$ is isomorphic to $T_gG$ so it should behave the same whether it acts on vector fields or on simple vectors (tangents at e). But regardless of how you take it, I still don't understand why there is a basis vector ($V_\mu$), if it acts only on vectors and not on one-forms.

 

[Silviu]

Every $(1,1)$ tensor, say $\varphi \otimes v \in U^* \otimes V$ is equivalent to the linear map $u \longmapsto \varphi(u)\cdot v$. Here we have $v \in T_gG$ and $\varphi \in T_e^*G$.

I am not sure I understand. Do you mean that a $(1,1)$ tensor can be transformed into a one-form by passing to it a one-form, which cancels the vector basis? But in our case, what would be that one-form? Like to me $\theta$ is clearly a $(1,1)$ tensor, and by the definition in the book the exactly same $\theta$, without any modification is called a one-form. I am not sure I understand it...

 

[martinbn]

Isn't a one-form by definition a function from vectors to real numbers?

Yes, but it is not a one-form, it is a $\mathfrak g$-valued one-form.

https://en.wikipedia.org/wiki/Vector-valued_differential_form