Exceptional Lie algebra explained

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:

ak{g} _{2,} ak{f} _{4,} ak{e} _{6,} ak{e} _{7,} ak{e} _{8} ; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are:

• 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 give a detailed construction of

ak{g} _{2} .

• Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
• Construct

ak{e} _{8} first and then find ak{e} _{6,} ak{e} _{7} as subalgebras.

• Tits has given a uniformed construction of the five exceptional Lie algebras.^[1]

References

• • Book: N. . Jacobson . Exceptional Lie Algebras . 1971 . 2017 . CRC Press . 978-1-351-44938-0.
• Book: Knapp . Anthony W. . Lie Groups Beyond an Introduction . 21 August 2002 . Springer Science & Business Media . 978-0-8176-4259-4 . en.

Further reading

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

Notes and References

1. Tits . Jacques . Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction . Indag. Math. . 1966 . 28 . 223–237 . 10.1016/S1385-7258(66)50028-2 . 9 August 2023.