dimension of standard representation of $\mathfrak{sl}_2(\Bbb C)$

Question

In my lecture notes I read that the standard representation of $\mathfrak{sl}_2(\Bbb C)$ is of dimension 2.

$\mathfrak{sl}_2(\Bbb C)$ is the set of matrices of $M_2(\Bbb C)$ of trace equal to 0. When embedded naturally into $M_2(\Bbb C)$, it should still be of dimension 3?

Thank you for your help.

Answer 1

That's the adjoint representation. The standard representation is the action of $\mathfrak{sl}(2,\mathbb C)$ in $\mathbb C^2$ defined by:$$(M,v)\mapsto M\cdot v.$$