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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras , especially the classification and representation theory of semisimple Lie algebras .
A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28. Spin(8)
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 Weyl group of the root system is the symmetry group of an equilateral triangle. Let be a root system in a Euclidean space .For each root , let denote the reflection about the hyperplane perpendicular to , which is given explicitly as
The center of SU(n) is isomorphic to the cyclic group /, and is composed of the diagonal matrices ζ I for ζ an n th root of unity and I the n × n identity matrix. Its outer automorphism group for n ≥ 3 is Z / 2 Z , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} while the outer automorphism group of SU(2) is the trivial group .
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their ...
The name derives from the fact that it is the root lattice of the E 8 root system. The norm [1] of the E 8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8.