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: ; 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 .
- Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
- Construct first and then find as subalgebras.
- Tits has given a uniformed construction of the five exception Lie algebras.[citation needed]
References[]
- ^ Fulton & Harris, Theorem 9.26.
- ^ Knapp, Appendix C, § 2.
- ^ 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.
- Jacobson, N. (2017) [1971]. Exceptional Lie Algebras. CRC Press. ISBN 978-1-351-44938-0.
Further reading[]
Categories:
- Lie algebras
- Exceptional Lie algebras
- Algebra stubs