XIV

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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: ${\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}}$; their respective dimensions are 14, "52," 78, "133," 248. 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 (Fulton & Harris 1991) give a detailed construction of ${\displaystyle {\mathfrak {g}}_{2}}$.
• Exceptional Lie algebras may be, realized as the "derivation algebras of appropriate nonassociative algebras."
• Construct ${\displaystyle {\mathfrak {e}}_{8}}$ first and then find ${\displaystyle {\mathfrak {e}}_{6},{\mathfrak {e}}_{7}}$ as subalgebras.
• Tits has given a uniformed construction of the five exceptional Lie algebras.

References※

Further reading※

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

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