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A subvariety of an affine algebraic variety $V\subseteq\mathbb{C}^n$ is an affine algebraic variety $W\subseteq\mathbb{C}^n$ that is contained in $V$.

So with respect to this definition, is it true that the set $U(n)$ of all unitary matrices is not an affine algebraic subvariety of $\mathbb{C}^{n^2}$?

Will be very happy to follow your comments. Thank you.

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up vote 1 down vote accepted

Indeed, it is not a subvariety.

One way to see it is to notice that $U(n)$ is a compact subset and that non-finite subvarieties of $\mathbb C^n$ are never compact.

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Related question here @Mariano: another trick we used a non-denumerable number of times? :) – Georges Elencwajg Apr 4 '12 at 20:50
Yeah... People are going to realise sooner or later! :D – Mariano Suárez-Alvarez Apr 4 '12 at 20:57
Anyway, I'll upvote your compact answer, Mariano ... – Georges Elencwajg Apr 4 '12 at 22:52

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