View source for Exceptional Lie algebra

{{Short description|Complex simple Lie Algebra}}
{{format footnotes |date=May 2024}}
In mathematics, an '''exceptional Lie algebra''' is a [[complex Lie algebra|complex]] [[simple Lie algebra]] whose [[Dynkin diagram]] is of exceptional (nonclassical) type.<ref>{{harvnb|Fulton|Harris|1991|loc=Theorem 9.26.}}</ref> There are exactly five of them: <math>\mathfrak{g}_2, \mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7, \mathfrak{e}_8</math>; their respective dimensions are 14, 52, 78, 133, 248.<ref name=Knapp>{{harvnb|Knapp|2002|loc=Appendix C, § 2.}}</ref> The corresponding diagrams are:<ref>{{harvnb|Fulton|Harris|1991|loc=§ 21.2.}}</ref>
*[[G2 (mathematics)|G<sub>2</sub>]] : {{Dynkin2|node|6a|node}}
*[[F4 (mathematics)|F<sub>4</sub>]] : {{Dynkin2|node|3|node|4a|node|3|node}}
*[[E6 (mathematics)|E<sub>6</sub>]] : {{Dynkin2|node|3|node|3|branch|3|node|3|node}}
*[[E7 (mathematics)|E<sub>7</sub>]] : {{Dynkin2|node|3|node|3|branch|3|node|3|node|3|node}}
*[[E8 (mathematics)|E<sub>8</sub>]] : {{Dynkin2|node|3|node|3|branch|3|node|3|node|3|node|3|node}}

In contrast, simple Lie algebras that are not exceptional are called [[classical Lie algebra]]s (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 {{harv|Fulton|Harris|1991}} give a detailed construction of <math>\mathfrak{g}_2</math>.
*Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
*Construct <math>\mathfrak{e}_8</math> first and then find <math>\mathfrak{e}_6, \mathfrak{e}_7</math> as subalgebras.
*[[Jacques Tits|Tits]] has given a uniformed construction of the five exceptional Lie algebras.<ref name="tits66">{{cite journal |last1=Tits |first1=Jacques |title=Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction |journal=Indag. Math. |date=1966 |volume=28 |pages=223–237 |doi=10.1016/S1385-7258(66)50028-2 |url=https://core.ac.uk/download/pdf/82411299.pdf |access-date=9 August 2023}}</ref>

== References ==
{{reflist}}
{{refbegin}}
* {{Fulton-Harris}}
* {{cite book |first1=N. |last1=Jacobson |title=Exceptional Lie Algebras |url=https://books.google.com/books?id=aLs6DwAAQBAJ |orig-date=1971 |date=2017 |publisher=CRC Press |isbn=978-1-351-44938-0}}
* {{cite book |last1=Knapp |first1=Anthony W. |title=Lie Groups Beyond an Introduction |date=21 August 2002 |publisher=Springer Science & Business Media |isbn=978-0-8176-4259-4 |url=https://books.google.com/books?id=U573NrppkA8C |language=en}}
{{refend}}

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

[[Category:Lie algebras]]
[[Category:Exceptional Lie algebras| ]]


{{algebra-stub}}