# 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]

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 exception Lie algebras.[citation needed]

## References

1. ^ Fulton & Harris, Theorem 9.26.
2. ^ Knapp, Appendix C, § 2.
3. ^ Fulton & Harris, § 21.2.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• N. Jacobson, "Exceptional Lie algebras" (1971)