In mathematics, an exceptional Lie algebra is a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
simple Lie algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.
A direct sum of si ...
whose
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
is of exceptional (nonclassical) type. There are exactly five of them:
; 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
.
*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.
References
*
*
Further reading
*https://www.encyclopediaofmath.org/index.php/Lie_algebra,_exceptional
*http://math.ucr.edu/home/baez/octonions/node13.html
Lie algebras
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