Exceptional Lie algebra
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: ; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are:
In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them).
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.
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