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How to Prove that

There is a closed non-zero $n$-form $\omega$ on $\text{GL}(n, \mathbb{R})$ which is left and right invariant.

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    You do mean $n$-form, not $n^2$-form, do you?2012-07-26
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    I mean $n$-form.2012-07-26
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    I've never heard of any conjugation invariant $n$-form $\omega_0$ on $M_n(\Bbb R)$ that is closed when prolonged to a left invariant $n$-form, i.e. that satisfies $\sum_{0\leq i for left-invariant vector fields on $\mathrm{GL}(n,\Bbb R)$.2012-07-26
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    $d(\log \det )$ comes to mind but I don't see how to get an $n$-form out of this.2012-07-27
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    @Mohammed: Why do you think the statement is true?2012-07-27

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