classification of indecomposable root systems

There are four infinite families of indecomposable root systems :

An ={±eiej:1i<jn+1};
Bn ={±ei±ej:1i<jn}{±ei:1in};
Cn ={±ei±ej:1i<jn}{±2ei:1in};
Dn ={±ei±ej:1i<jn}

The subscript on the name of the root system is the dimensionPlanetmathPlanetmathPlanetmath of 𝐄, the ambient Euclidean space containing the root system. In the case of An, the ambient 𝐄 is the n-dimensional subspacePlanetmathPlanetmathPlanetmath perpendicularMathworldPlanetmathPlanetmathPlanetmath to i=1nei. In the other 3 cases, 𝐄=n. Throughout, we endow n with the standard Euclidean inner product, and let ei, 1in 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(i=16(±ei)±(e7-e8)):4 minus signs};
E7 =A8{i=18(±ei):4 minus signs};
E8 =D8{i=18(±ei):even number of minus signs}.

The following table indicates the cardinality of and the Lie algebrasMathworldPlanetmath and Dynkin diagramsMathworldPlanetmath corresponding to the above root systems.

Figure 1: Irreducible root systems and simple Lie algebrasMathworldPlanetmath.
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