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: g 2 , f 4 , e 6 , e 7 , e 8 {\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).
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:
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