Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
F4 (mathématiques)
Résumé
DISPLAYTITLE:F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.
The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.
There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.
The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8.
In older books and papers, F4 is sometimes denoted by E4.
The Dynkin diagram for F4 is: .
Its Weyl/Coxeter group G = W(F4) is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree μ(G) = 24, which is realized by the action on the 24-cell.
The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin.
The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations, representing the vertices of a disphenoidal 288-cell if the edge lengths of the 24-cells are equal:
24-cell vertices:
24 roots by (±1, ±1, 0, 0), permuting coordinate positions
Dual 24-cell vertices:
8 roots by (±1, 0, 0, 0), permuting coordinate positions
16 roots by (±1/2, ±1/2, ±1/2, ±1/2).
One choice of simple roots for F4, , is given by the rows of the following matrix:
Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ...
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.