In mathematics, F₄ is the name of a Lie group and also its Lie algebra . 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, a fundamental representation is a representation of a mathematical structure, such as a group, that satisfies the following condition: All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself. ...

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. In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Hans Freudenthal (September 17, 1905 – October 13, 1990) was a Dutch mathematician born in Luckenwalde in Germany into a Jewish family. ... Jacques Tits (born August 12, 1930) is a French mathematician, formerly Belgian. ...

There are 3 real forms: a compact one, a split one, and a third one. In mathematics, the simple Lie groups were classified by Élie Cartan. ...

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. To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, E8 is the name of a root system and of several associated Lie groups and also their Lie algebras . ...

Algebra

Dynkin diagram

See also Simple Lie group. ... Image File history File links Dynkin_diagram_F4. ...

Roots of F₄

Simple roots

(0,0,0,1)

(0,0,1, − 1)

(0,1, − 1,0)

Weyl/Coxeter group

Its Weyl/Coxeter group is the symmetry group of the 24-cell. In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ... The symmetry group of an object (e. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ...

Cartan matrix

In mathematics, the term Cartan matrix has two meanings. ...

F₄ lattice

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 24-cell. See lattice for other meanings of this term, both within and without mathematics. ... In crystallography, the cubic crystal system is the most symmetric of the 7 crystal systems. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (a mixture of integers and half-integers is not allowed). ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ...

References

• John Baez, The Octonions, Section 4.2: F₄, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at

http://math.ucr.edu/home/baez/octonions/node15.html. John Carlos Baez (b. ...