Exceptional Lie algebra

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type.^[1] There are exactly five of them: ${\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}}$; their respective dimensions are 14, 52, 78, 133, 248.^[2] The corresponding diagrams are:^[3]

• G₂ :
• F₄ :
• E₆ :
• E₇ :
• E₈ :

In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them).

Construction

There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions:

• § 22.1-2 of (Fulton & Harris 1991) give a detailed construction of ${\displaystyle {\mathfrak {g}}_{2}}$.
• Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
• Construct ${\displaystyle {\mathfrak {e}}_{8}}$ first and then find ${\displaystyle {\mathfrak {e}}_{6},{\mathfrak {e}}_{7}}$ as subalgebras.
• Tits has given a uniformed construction of the five exceptional Lie algebras.^[4]

References

1. Fulton & Harris 1991, Theorem 9.26.
2. Knapp 2002, Appendix C, § 2.
3. Fulton & Harris 1991, § 21.2.
4. Tits, Jacques (1966). "Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction" (PDF). Indag. Math. 28: 223–237. doi:10.1016/S1385-7258(66)50028-2. Retrieved 9 August 2023.

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Jacobson, N. (2017) [1971]. Exceptional Lie Algebras. CRC Press. ISBN 978-1-351-44938-0.
• Knapp, Anthony W. (21 August 2002). Lie Groups Beyond an Introduction. Springer Science & Business Media. ISBN 978-0-8176-4259-4.

Further reading

• https://www.encyclopediaofmath.org/index.php/Lie_algebra,_exceptional
• http://math.ucr.edu/home/baez/octonions/node13.html