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Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M_n(F)×M_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.

How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?

Thank you.

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Show that $\mathrm{GL}_n(F)$ is dense in $\mathrm{GL}_n(K) \subset K^{n^2+1}$ for the Zariski topology. –  Plop May 6 '11 at 23:20
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1 Answer 1

I believe that $N=M$, since if we consider two generic matrices $X$ and $Y$ in variables $x_{ij}$ and $y_{ij}$, respectively, then the equation $[X,Y] = 0$ translates into a set of polynomial equations in the aforementioned $2 n^2$ variables. Let $I$ be the ideal generated by the above polynomials in the ring of polynomials in $2 n^2$ variables. Then $M = Z(I)$. Now the second statement follows easily from the fact that for every $A \in GL(n,K)$ we have $[AXA^{-1},AYA^{-1}] = A[X,Y]A^{-1}$.

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