Exceptional Lie algebra e6

The exceptional complex Lie algebra $\mathfrak e_6$ is the unique simple complex Lie algebra whose root system has Dynkin type $E_6$ . It has complex dimension $78$ , rank $6$ , and $72$ roots.

Its two dual $27$ -dimensional minuscule modules $\mathbf{27}$ and $\mathbf{27}^*$ are its smallest nontrivial irreducible representations. Its adjoint representation is $\mathbf{78}$ .

Jordan-algebra realization

Let $J_{\mathbb C}=H_3(\mathbb O)\otimes_{\mathbb R}\mathbb C$ be the complexified exceptional Jordan algebra. It is $27$ -dimensional and carries a cubic determinant. The Lie algebra $\mathfrak e_6$ can be realized as the infinitesimal determinant-preserving linear transformations of $J_{\mathbb C}$ . This realizes $J_{\mathbb C}$ as a $\mathbf{27}$ , and its trace dual as $\mathbf{27}^*$ .

Groups and real forms

The simply connected complex group of type $E_6$ has center $\mu_3$ ; its adjoint quotient is centerless. The compact simply connected real form is compact $E_6$ . Other real forms include the split form $E_{6(6)}$ and the real form $E_{6(-26)}$ acting as determinant-preserving transformations of the real Albert algebra. These groups share a complexified Lie algebra but are not interchangeable.

Branching and paper context

Deleting an end node from the $E_6$ Dynkin diagram gives a regular subalgebra $\mathfrak{so}_{10}\oplus\mathbb C$ . With one common charge normalization,

\mathbf{27}\cong\mathbf1_{4}\oplus\mathbf{10}_{-2}\oplus\mathbf{16}_{1},

and reversing all charges gives an equivalent convention. This is one reason $E_6$ is prominent in grand-unified representation theory.

In the three-generation construction, $\mathfrak e_6$ occurs in the regular chain

\mathfrak{sl}_3\oplus\mathfrak{sl}_2 \subset\mathfrak{sl}_5\subset\mathfrak{so}_{10} \subset\mathfrak e_6\subset\mathfrak e_7.

This chain is summarized by the $E_n$ convention.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plate V. Publisher record.
John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 6--8. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, Chapter 6. Publisher record.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.