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

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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 exceptional Lie algebras.[4]

References

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  1. ^ Fulton & Harris 1991, Theorem 9.26.
  2. ^ Knapp 2002, Appendix C, § 2.
  3. ^ Fulton & Harris 1991, § 21.2.
  4. ^ Tits, Jacques (1966). "Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction" (PDF). Indag. Math. 28: 223–237. doi:10.1016/S1385-7258(66)50028-2. Retrieved 9 August 2023.

Further reading

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