Special linear Lie algebra

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

\mathfrak{sl}_n(\mathbb F)=\{X\in M_n(\mathbb F): \mathrm{tr}(X)=0\},

equipped with the commutator bracket

[X,Y]=XY-YX

(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).

Dimensions and scalar restriction

One has

\dim_{\mathbb R}\mathfrak{sl}_n(\mathbb R)=n^2-1,\qquad \dim_{\mathbb C}\mathfrak{sl}_n(\mathbb C)=n^2-1.

The underlying real Lie algebra $\mathfrak{sl}_n(\mathbb C)_{\mathbb R}$ therefore has real dimension $2(n^2-1)$ . In particular,

\dim_{\mathbb C}\mathfrak{sl}_2(\mathbb C)=3,\qquad \dim_{\mathbb R}\mathfrak{sl}_2(\mathbb C)_{\mathbb R}=6.

The latter is the Lie algebra of both $SL(2,\mathbb C)_{\mathbb R}$ and, after quotienting by the discrete center, $PSL(2,\mathbb C)_{\mathbb R}$ . It is isomorphic as a real Lie algebra to $\mathfrak{so}(1,3)$ .

References

Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 2–3. Publisher record.
Jean-Pierre Serre, Complex Semisimple Lie Algebras, Springer, 1987, Chapter I. Publisher record.