Complex Lie algebra sl3(C)

The complex Lie algebra $\mathfrak{sl}_3(\mathbb C)$ is the algebra of trace-zero $3\times3$ complex matrices with commutator bracket. It is simple, has complex dimension $8$ , rank $2$ , and Dynkin type $A_2$ .

Its defining module is $\mathbf 3=\mathbb C^3$ . The other nontrivial fundamental representation is the dual $\mathbf{\bar 3}=(\mathbb C^3)^*$ , and the adjoint representation is the eight-dimensional summand in

\mathbf 3\otimes\mathbf{\bar 3}\cong \mathbf 1\oplus\mathbf 8.

Roots and weights

For the Cartan subalgebra of diagonal trace-zero matrices, write $\varepsilon_i$ for the $i$ -th diagonal coordinate. The roots are

\varepsilon_i-\varepsilon_j\qquad(i\ne j),

and one may take $\varepsilon_1-\varepsilon_2$ and $\varepsilon_2-\varepsilon_3$ as simple roots. The corresponding Dynkin diagram has two nodes joined by one edge.

The tensor products $\mathbf3\otimes\mathbf3\cong\mathbf6\oplus\mathbf{\bar3}$ and $\mathbf3\otimes\mathbf{\bar3}\cong\mathbf1\oplus\mathbf8$ are frequently used to recognize $A_2$ -modules.

Groups and real forms

The simply connected complex group is $SL(3,\mathbb C)$ , with center $\mu_3$ ; its adjoint form is $PSL(3,\mathbb C)$ . The compact real form integrates to $SU(3)$ , while $SL(3,\mathbb R)$ is the split real form. Thus a paper that writes $\mathfrak{sl}_3$ over $\mathbb C$ is not automatically referring to the compact group $SU(3)$ , even though their representation theories are closely related by complexification.

Paper context

In the three-generation construction, one copy is the color factor in the complexified Standard Model algebra $\mathbb C\oplus\mathfrak{sl}_2\oplus\mathfrak{sl}_3$ , while a commuting copy $\mathfrak{sl}_3^{\mathrm{gen}}$ relates the three generation subspaces. In the exceptional-Jordan-algebra construction, the compact group $SU(3)$ is the automorphism group of the octonions that fixes a chosen complex subalgebra; its defining $\mathbb C^3$ module appears in $\mathbb O\cong\mathbb C\oplus\mathbb C^3$ .

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, Sections 8--13. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Sections 12--13. Publisher record.
John C. Baez and Paul Schwahn, The Standard Model Gauge Group from the Exceptional Jordan Algebra, 2026. arXiv:2606.15235.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.