Standard sl5 in e7 - Knowlpedia

Statement

For a good embedded Standard Model Lie algebra and its standard $\mathfrak{sl}_6$ , there is a unique Lie subalgebra $\mathfrak s\cong\mathfrak{sl}_5(\mathbb C)$ such that

\mathfrak g_{\mathrm{SM}} \subset\mathfrak s \subset\mathfrak{sl}_6^{\mathrm{SM}}.

This $\mathfrak s$ is regular in $\mathfrak e_7$ and is called the standard $\mathfrak{sl}_5$ , denoted $\mathfrak{sl}_5^{\mathrm{SM}}$ .

All displayed containments are Lie-subalgebra inclusions. In particular,

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

is a chain, not a direct-sum decomposition.

Root-system construction

Let $\mathfrak h_{\mathrm{SM}}$ be the Cartan subalgebra of the regular $\mathfrak g_{\mathrm{SM}}$ , and let $U$ be its real form in the $E_7$ root space. The roots in $\Phi(E_7)\cap U$ form an $A_4$ subsystem of $20$ roots, hence define a regular copy of $\mathfrak{sl}_5$ . Since $U\perp P$ , this subsystem lies in the $A_5$ roots of $\mathfrak{sl}_6^{\mathrm{SM}}$ .

Why uniqueness does not require regularity

View the defining module of $\mathfrak{sl}_6^{\mathrm{SM}}$ as $\mathbb C^6$ . Any embedded $\mathfrak{sl}_5\subset\mathfrak{sl}_6$ acts as

\mathbb C^6=W\oplus L, \qquad \dim W=5, \qquad \dim L=1,

with $W$ the defining module and $L$ trivial. Under $\mathfrak g_{\mathrm{SM}}$ , the fixed-vector subspace of the defining $\mathbb C^6$ is exactly one line, so $L$ is forced. The remaining irreducible $3$ - and $2$ -dimensional Standard Model summands admit no equivariant map to $L$ , so their invariant complement $W$ , and hence the embedded $\mathfrak{sl}(W)$ , are forced as well.

Physical representation-theory role

This is the Lie-algebra form of the Georgi–Glashow $SU(5)$ embedding. Restricting the generation modules through $\mathfrak{sl}_6^{\mathrm{SM}}\supset\mathfrak{sl}_5^{\mathrm{SM}}\supset\mathfrak g_{\mathrm{SM}}$ produces the exterior-algebra model of one Standard Model generation.

References

John C. Baez, “Three Generations in E7,” 2026, Proposition 8. arXiv:2608.06271.
Howard Georgi and Sheldon L. Glashow, “Unity of All Elementary-Particle Forces,” Physical Review Letters 32 (1974), 438–441. DOI record.
John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv:0904.1556.