Exceptional Lie algebra e7

The exceptional complex Lie algebra $\mathfrak e_7$ is the unique simple complex Lie algebra whose root system has Dynkin type $E_7$ . It has complex dimension $133$ , rank $7$ , and $126$ roots.

Its smallest nontrivial irreducible representation is the $56$ -dimensional minuscule module $\mathbf{56}$ , which admits a nondegenerate invariant alternating form. Its adjoint module is $\mathbf{133}$ .

Distinguished structure and subalgebras

The $\mathbf{56}$ may be modeled using the Freudenthal space attached to the exceptional Jordan algebra; in addition to its alternating form it carries an invariant quartic form. Infinitesimal transformations preserving this structure give $\mathfrak e_7$ .

Three useful maximal-rank subalgebras and adjoint branchings are

\begin{aligned} \mathfrak e_7&\supset\mathfrak{sl}_8, &\mathbf{133}&=\mathbf{63}\oplus\mathbf{70},\\ \mathfrak e_7&\supset\mathfrak{sl}_2\oplus\mathfrak{so}_{12}, &\mathbf{133}&=(\mathbf3,\mathbf1)\oplus(\mathbf1,\mathbf{66})\oplus(\mathbf2,\mathbf{32}),\\ \mathfrak e_7&\supset\mathfrak{sl}_3\oplus\mathfrak{sl}_6, &\mathbf{133}&=(\mathbf8,\mathbf1)\oplus(\mathbf1,\mathbf{35}) \oplus(\mathbf3,\mathbf{15})\oplus(\mathbf{\bar3},\mathbf{15}^*). \end{aligned}

Here $\mathbf{15}=\Lambda^2\mathbb C^6$ and $\mathbf{15}^*=\Lambda^4\mathbb C^6$ . The displayed summands have dimensions $8+35+45+45=133$ .

Groups and real forms

The simply connected complex group of type $E_7$ has center of order $2$ , acting nontrivially on $\mathbf{56}$ ; the adjoint complex group is its centerless quotient. The compact simply connected real form is compact $E_7$ , and the split real form is commonly denoted $E_{7(7)}$ . Statements about the complex Lie algebra alone do not select one of these global or real forms.

Three generations

The three-generation construction begins with an embedded complexified Standard Model algebra

\mathfrak g_{\mathrm{SM}}=\mathbb C\oplus\mathfrak{sl}_2\oplus\mathfrak{sl}_3 \subset\mathfrak e_7.

It identifies a commuting generation-symmetry $\mathfrak{sl}_3$ , three associated $\mathfrak{sl}_2\oplus\mathfrak{so}_{12}$ subalgebras, and a vector-space decomposition

\mathfrak e_7\cong(\mathfrak{sl}_6\oplus\mathbb C^2) \oplus V_1\oplus V_2\oplus V_3, \qquad \dim V_i=32.

Each $V_i$ , under the adjoint action restricted to $\mathfrak g_{\mathrm{SM}}$ , is equivalent to the action on $\Lambda\mathbb C^5$ describing one generation of fermions and antifermions, including a right-handed neutrino and its antiparticle.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plate VI. Publisher record.
John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 7--9. Publisher record.
Hans Freudenthal, "Beziehungen der $E_7$ und $E_8$ zur Oktavenebene. I," Indagationes Mathematicae 16 (1954), 218--230. DOI50031-X).
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.