Special Linear Lie Algebra
Definition
The set of matrices whose trace is $0$ is called $\mathfrak{sl}(n, \mathbb{C})$.
$$ \mathfrak{sl}(n, \mathbb{C}) = \left\{X \in M_{n}(\mathbb{C}) : \operatorname{tr}(X) = 0 \right\} $$
Explanation
If the bracket with respect to matrix multiplication is defined as $[X, Y] = XY - XY$, then $\mathfrak{sl}(n, \mathbb{C})$ forms a Lie algebra.
Judging from the definition alone, it does not appear to be related to the special linear group $\operatorname{SL}(n, \mathbb{C})$, but in fact it becomes the Lie algebra of $\operatorname{SL}$.
Properties
Dimension
Since $\mathfrak{sl}(n, \mathbb{C})$ has one added linear constraint $x_{11} + x_{22} + \cdots + x_{nn} = 0$ that the trace of $X$ is $0$, its dimension is one less than that of the general linear Lie algebra $\mathfrak{gl}(n, \mathbb{C}) = M_{n}(\mathbb{C})$.
$$ \dim \mathfrak{sl}(n, \mathbb{C}) = n^{2} - 1 $$