Complex Lie algebra sl6(C)

The complex Lie algebra $\mathfrak{sl}_6(\mathbb C)$ is the algebra of trace-zero $6\times6$ complex matrices with commutator bracket. It is simple, has complex dimension $35$ , rank $5$ , and Dynkin type $A_5$ .

Its defining module is $\mathbf6=\mathbb C^6$ . The fundamental modules are $\Lambda^k\mathbb C^6$ for $1\leq k\leq5$ , of dimensions $6,15,20,15,6$ . In particular $\Lambda^3\mathbb C^6$ is a self-dual $20$ -dimensional module, while the adjoint module has dimension $35$ .

Root data and groups

The roots are $\varepsilon_i-\varepsilon_j$ , and the five consecutive differences form a simple system of type $A_5$ . The simply connected complex group is $SL(6,\mathbb C)$ , with center $\mu_6$ , and its compact real form is $SU(6)$ . Neither should be identified with the adjoint group $PSL(6,\mathbb C)$ , even though all have the expected closely related Lie algebras.

Paper context

In the three-generation construction, a distinguished $\mathfrak{sl}_6^{\mathrm{SM}}$ is the centralizer of the generation-symmetry $\mathfrak{sl}_3^{\mathrm{gen}}$ inside $\mathfrak e_7$ . Together they form a maximal-rank regular subalgebra. The corresponding branching of the adjoint module is

\mathbf{133} \cong(\mathbf{35},\mathbf1)\oplus(\mathbf1,\mathbf8) \oplus(\mathbf{15},\mathbf3)\oplus(\mathbf{15}^*,\mathbf{\bar3})

under $\mathfrak{sl}_6\oplus\mathfrak{sl}_3$ , where $\mathbf{15}=\Lambda^2\mathbb C^6$ and $\mathbf{15}^*=\Lambda^4\mathbb C^6$ .

The paper also uses the common intersection $\mathfrak{sl}_6\oplus\mathbb C^2$ of three $\mathfrak{sl}_2\oplus\mathfrak{so}_{12}$ subalgebras and a chain through the complex Lie algebra $\mathfrak{sl}_5$ :

\mathfrak g_{\mathrm{SM}}\subset\mathfrak{sl}_5 \subset\mathfrak{sl}_6\subset\mathfrak{sl}_6\oplus\mathbb C^2 \subset\mathfrak e_7.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plates I and VI. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Section 15. Publisher record.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.