Complex Lie algebra sl5(C)

The complex Lie algebra $\mathfrak{sl}_5(\mathbb C)$ consists of trace-zero $5\times5$ complex matrices with commutator bracket. It is simple, of complex dimension $24$ , rank $4$ , and Dynkin type $A_4$ .

Its four fundamental representations are the exterior powers

\mathbb C^5,\quad \Lambda^2\mathbb C^5,\quad \Lambda^3\mathbb C^5,\quad\Lambda^4\mathbb C^5,

of dimensions $5,10,10,5$ . After choosing a volume form, the last two are dual to the first two. The adjoint representation has dimension $24$ .

Root data

For the diagonal Cartan subalgebra, the roots are $\varepsilon_i-\varepsilon_j$ for $i\ne j$ . The simple roots $\varepsilon_i-\varepsilon_{i+1}$ , $1\leq i\leq4$ , give the four-node chain $A_4$ . The Weyl group is the symmetric group $S_5$ , acting by permuting the diagonal coordinates.

Groups and real forms

The simply connected complex group is $SL(5,\mathbb C)$ , whose center is $\mu_5$ ; its adjoint quotient is $PSL(5,\mathbb C)$ . The compact real form integrates to $SU(5)$ . The finite-dimensional complex representations of the compact and complex forms share the same highest-weight classification, but the groups and their real Lie algebras are different objects.

Paper context

The block-diagonal subgroup

S(U(2)\times U(3))\subset SU(5)

is isomorphic to $(U(1)\times SU(2)\times SU(3))/\mathbb Z_6$ . Restricting the $32$ -dimensional exterior algebra $\Lambda\mathbb C^5$ to this subgroup gives one Standard Model generation together with antiparticles and a right-handed neutrino.

In the three-generation construction, a unique compatible copy of $\mathfrak{sl}_5$ fits into a chain through the complex Lie algebra $\mathfrak{sl}_6$ and the exceptional Lie algebra $\mathfrak e_7$ :

\mathfrak g_{\mathrm{SM}}\subset\mathfrak{sl}_5 \subset\mathfrak{sl}_6 \subset\mathfrak e_7.

This is also the $A_4$ term of the regular-subalgebra chain summarized by the $E_n$ series.

References

William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Sections 13 and 15. Publisher record.
John C. Baez and John Huerta, The Algebra of Grand Unified Theories, Bulletin of the American Mathematical Society 47 (2010), 483--552. DOI.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.
John C. Baez and Paul Schwahn, The Standard Model Gauge Group from the Exceptional Jordan Algebra, 2026. arXiv:2606.15235.