Special linear Lie algebra

For $\mathbb F\in\{\mathbb R,\mathbb C\}$, the special linear Lie algebra is

equipped with the commutator bracket

(see Lie bracket and compare general linear Lie algebra ).

This Lie algebra is the Lie algebra of the special linear group $SL(n,\mathbb F)$: under the identification $T_I SL(n,\mathbb F)\cong \mathfrak{sl}_n(\mathbb F)$, tangent vectors at the identity are exactly the trace-zero directions. Equivalently, $\mathfrak{sl}_n(\mathbb F)$ is the kernel of the differential of $\det:GL(n,\mathbb F)\to\mathbb F^\times$ at the identity.

Over $\mathbb C$, $\mathfrak{sl}_n(\mathbb C)$ is a fundamental example of a complex simple Lie algebra for $n\ge 2$ (see simple Lie algebra and classification of simple Lie algebras ). The case $n=2$ is the standard testbed for root computations (see standard sl2 example ).