Compact exceptional Lie group E8

The compact exceptional Lie group $E_8$ is the compact connected simple Lie group with root system of Dynkin type $E_8$ . It has rank $8$ and real dimension $248$ . It is both simply connected and centerless, so its simply connected and adjoint forms coincide. Its Lie algebra is the compact real form whose complexification is $\mathfrak e_8$ .

The smallest nontrivial complex representation is the complexified adjoint representation, of dimension $248$ . The absence of a nontrivial center agrees with the equality of the $E_8$ root and weight lattices.

Distinguishing the forms

Compact $E_8$ is not the complex group $E_8(\mathbb C)$ and not the split real group $E_{8(8)}$ . They share the complex Lie algebra $\mathfrak e_8$ , but compact $E_8$ has negative-definite Killing form on its real Lie algebra and every finite-dimensional complex representation is unitarizable.

Unlike types $E_6$ and $E_7$ , there is no ambiguity between simply connected and adjoint compact forms for $E_8$ . There can still be finite central quotients in descriptions of its proper connected subgroups.

Paper context

Neither construction uses compact $E_8$ as its ambient symmetry group: the three-generation construction uses complex $\mathfrak e_7$ , and the exceptional-Jordan-algebra construction uses compact $F_4$ . Compact $E_8$ belongs to the wider $E_n$ family and provides context for why extending an $E_7$ construction is mathematically tempting. Such an extension does not follow merely from the inclusion of Dynkin diagrams, and known representation-theoretic obstructions apply.

References

John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 8--10. Publisher record.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhauser, 2002, Chapter VII. Publisher record.
Jacques Distler and Skip Garibaldi, "There is no 'Theory of Everything' inside $E_8$ ," Communications in Mathematical Physics 298 (2010), 419--436. DOI.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.