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## Homework Statement

Show that

[tex]d\omega_{ij}+\sum_{k=1}^{n} \omega_{ik}\wedge\omega_{kj} =0[/tex]

## Homework Equations

Let G be the group of invertible

*n*x

*n*matrices. This is an open set in the vector space

[tex]M=Mat(n\times n, R)[/tex]

and our formalism of differential forms applies there with the coordinate functions now being the entries

[tex]X_{ij}[/tex] of a matrix [tex]X[/tex].

For every linear function

[tex]\lambda : M\rightarrow R[/tex],

there is a unique 1-form

[tex]\omega_{\lambda}\in G[/tex]

that has the value[tex]\lambda[/tex] at

*I*and is invariant under all left multiplications

[tex]L_{\lambda}:Y\rightarrow YX for X\in G[/tex]; that is, [tex]L_{X}^{*}(\omega_{\lambda})=\omega_{\lambda}[/tex]. It is given by

[tex]\omega_{\lambda}(X):Y|\rightarrow\lambda(D(L_{X^{-1}})_{X}(Y))=\lambda(X^{-1}Y), X\in G, Y\in M[/tex].

The basic examples are the Maurer-Cartan forms

[tex]\omega_{ij}(X)(Y)=(X^{-1}Y)_{ij}[/tex]

or

[tex]\omega_{ij}(X)=\sum_{k=1}^{n} (X^{-1}_{ik}dX_{kj})[/tex].

## The Attempt at a Solution

HA HA, I don't know. It has been suggested that I use an equivalent formula, namely

[tex]dX_{ij}=\sum_{k=1}^{n} X_{ik}\omega_{kj}[/tex].

Then apply [tex]d[/tex]. Now this will hold for all

*i*and

*j*. To obtain a differential [tex]d\omega_{rj}[/tex], multiply that equation by [tex](X^{-1})_{ri}[/tex] and sum over

*i*.

This suggestion is great, but I am not even sure I really understand it. Any other suggestions on how to prove this, or maybe some tips about this suggestion?

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