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There is a problem given in a representation theory textbook:

Prove that for any finite-dimensional complex vector space $V$ there are no $X, Y \in \operatorname{End}V$ such that $[X, Y] = \mathrm{id}$.

I tried looking at $\mathbb{C}[X, Y]$ and the ideal of $\operatorname{End}V$ generated by $XY - YX$, but so far to no avail. I could use a hint.

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    @KCd, awesome! I went through the case $(\mathbb{Z}/2)^2$ by hand, and it's great to know how deep this problem really is!2012-05-19

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