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The rank of a root system Φ is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A 2, B 2, and G 2 pictured to the right, is said to be irreducible.
The central classification is that a simple Lie algebra has a root system, to which is associated an (oriented) Dynkin diagram; all three of these may be referred to as B n, for instance. The unoriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, which is the finite reflection group associated to the root system.
The shaded region is the fundamental Weyl chamber for the base {,}. If is a root system, we may consider the hyperplane perpendicular to each root .Recall that denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of generated by all the 's.
This image is a derivative work of the following images: File:Root-system-G2.png licensed with Cc-by-sa-3.0-migrated, GFDL . 2006-03-20T17:01:21Z Maksim 582x607 (6784 Bytes) La bildo estas kopiita de wikipedia:en.
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An alternative (7-dimensional) description of the root system, which is useful in considering E 7 × SU(2) as a subgroup of E 8, is the following: All () permutations of (±1,±1,0,0,0,0,0) preserving the zero at the last entry, all of the following roots with an even number of + 1 / 2
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