Compact exceptional Lie group E7

The compact exceptional Lie group $E_7$ means here the compact, connected, simply connected simple Lie group with root system of Dynkin type $E_7$ . It has rank $7$ , real dimension $133$ , and center isomorphic to $\mathbb Z/2\mathbb Z$ . Its Lie algebra is the compact real form whose complexification is $\mathfrak e_7$ .

Its smallest nontrivial complex representation is the faithful $56$ -dimensional module $\mathbf{56}$ , which preserves a nondegenerate alternating form. The adjoint representation has dimension $133$ and factors through the centerless adjoint quotient $E_7/(\mathbb Z/2\mathbb Z)$ .

Global and real-form caveat

Some sources use “compact $E_7$ ” for the adjoint group. The simply connected and adjoint forms have the same Lie algebra, dimension, and local geometry, but the $\mathbf{56}$ does not descend to the adjoint form. This knowl consistently uses the simply connected form.

Compact $E_7$ is not the complex Lie group $E_7(\mathbb C)$ , nor a noncompact real form such as split $E_{7(7)}$ . Statements proved by complex root-space calculations must be translated carefully when global topology or disconnected stabilizers matter.

Translation of the three-generation construction

The three-generation construction performs its calculations in the complex Lie algebra $\mathfrak e_7$ , but it can be translated to the compact real form and the compact Standard Model gauge group

S(U(2)\times U(3))\cong (U(1)\times SU(2)\times SU(3))/\mathbb Z_6.

At group level the subgroups integrating $\mathfrak{sl}_2\oplus\mathfrak{so}_{12}$ or $\mathfrak{sl}_3\oplus\mathfrak{sl}_6$ involve finite central quotients; the displayed Lie-algebra direct sums alone do not determine those quotients.

References

John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 7--9. Publisher record.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhauser, 2002, Chapter VII. Publisher record.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.