classification of indecomposable root systems
There are four infinite families of indecomposable root systems :
An | ={±ei∓ej:1≤i<j≤n+1}; | ||
Bn | ={±ei±ej:1≤i<j≤n}∪{±ei:1≤i≤n}; | ||
Cn | ={±ei±ej:1≤i<j≤n}∪{±2ei:1≤i≤n}; | ||
Dn | ={±ei±ej:1≤i<j≤n} |
The subscript on the name of the root system is the dimension of 𝐄,
the ambient Euclidean space containing the root system. In the case of
An, the ambient 𝐄 is the n-dimensional subspace
perpendicular
to ∑ni=1ei. In the other 3 cases, 𝐄=ℝn.
Throughout, we endow ℝn with the standard Euclidean inner
product, and let ei, 1≤i≤n denote the standard basis.
As well, there are 5 exceptional, crystallographic root systems:
G2 | =A3∪{±13(2e1-e2-e3),±13(-e1+2e2-e3),±13(-e1-e2+2e3)}; | ||
F4 | =B4∪{12(±e1±e2±e3±e4)}; | ||
E6 | =A6∪{±(e7-e8)}∪{12(6∑i=1(±ei)±(e7-e8)):4 minus signs}; | ||
E7 | =A8∪{8∑i=1(±ei):4 minus signs}; | ||
E8 | =D8∪{8∑i=1(±ei):even number of minus signs}. |
The following table indicates the cardinality of and the Lie algebras
and Dynkin diagrams
corresponding to the above root systems.

Title | classification of indecomposable root systems |
---|---|
Canonical name | ClassificationOfIndecomposableRootSystems |
Date of creation | 2013-03-22 15:28:56 |
Last modified on | 2013-03-22 15:28:56 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 6 |
Author | rmilson (146) |
Entry type | Result |
Classification | msc 17B20 |