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  2. Root system - Wikipedia

    en.wikipedia.org/wiki/Root_system

    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 .

  3. E8 (mathematics) - Wikipedia

    en.wikipedia.org/wiki/E8_(mathematics)

    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.

  4. SO(8) - Wikipedia

    en.wikipedia.org/wiki/SO(8)

    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)

  5. Dynkin diagram - Wikipedia

    en.wikipedia.org/wiki/Dynkin_diagram

    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.

  6. Weyl group - Wikipedia

    en.wikipedia.org/wiki/Weyl_group

    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

  7. Special unitary group - Wikipedia

    en.wikipedia.org/wiki/Special_unitary_group

    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 .

  8. Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Euclidean_space

    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 ...

  9. E8 lattice - Wikipedia

    en.wikipedia.org/wiki/E8_lattice

    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.