Let $\mathfrak{sl}(n,\mathbb{R})=\{X\in\mathfrak{gl}(n,\mathbb{R}): \mathrm{tr}(X)=0\}$ the Lie algebra of $\mathrm{SL}(n,\mathbb{R})$, i want show that, if $X\in\mathfrak{sl}(n,\mathbb{R})$ is such that: $[X,Y]=0$, for all $Y\in\mathfrak{sl}(n,\mathbb{R})$, then: $X=0$. Where, for all $A,B\in\mathfrak{sl}(n,\mathbb{R})$: $$[A,B]=AB-BA.$$ I have no idea how to proceed. Any help would be appreciated!
Asked
Active
Viewed 75 times
0
-
Take for $Y$ a lot of matrices and see what happens. – Rafael Mrden Sep 27 '19 at 14:50
-
@RafaelMrđen i don't understand your answer. – inoc Sep 27 '19 at 15:08
-
For example, try $Y = E_{ij}$, for $i\ne j$, the matrix with a $1$ in the $i$th row and $j$th column, $0$'s everywhere else. – Ted Shifrin Sep 27 '19 at 16:45
-
4Possible duplicate of Centre of Lie Algebra $sl_n(\mathbb{F})$ – Dietrich Burde Sep 27 '19 at 18:07
-
Since $\mathfrak{sl}(n,\mathbb{R})$ is simple and the centre $Z$ is an ideal, it must be zero or $\mathfrak{sl}(n,\mathbb{R})$. Since the Lie algebra is not abelian, it follows that $Z=0$. – Dietrich Burde Sep 27 '19 at 18:09