mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, F₄ is the name of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...

and also its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

f₄. It is one of the five exceptional

simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...

s. F₄ has rank 4 and dimension 52. The compact form is simply connected and its

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...

is the

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...

. Its

fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...

is 26-dimensional. The compact real form of F₄ is the

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...

of a 16-dimensional

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...

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 Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life an ...

. 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 algebra In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there ...

s. The F₄ Lie algebra may be constructed by adding 16 generators transforming as a

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

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.

Algebra

Dynkin diagram

The

Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...

for F₄ is: .

Weyl/Coxeter group

Its

Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

group is the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of the

24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...

: it is a

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...

of order 1152. It has minimal faithful degree , which is realized by the action on the

Cartan matrix

\left \begin
2&-1&0&0\\
-1&2&-2&0\\
0&-1&2&-1\\
0&0&-1&2
\end \right /math>

F₄ lattice

The F₄ lattice is a four-dimensional

body-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...

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

centered at the origin.

Roots of F₄

The 48 root vectors of F₄ can be found as the vertices of the

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).

Simple roots

One choice of simple roots for F₄, , is given by the rows of the following matrix: :

\begin
0&1&-1&0 \\
0&0&1&-1 \\
0&0&0&1 \\
\frac&-\frac&-\frac&-\frac\\
\end

F₄ polynomial invariant

Just as O(''n'') is the group of automorphisms which keep the quadratic polynomials invariant, F₄ is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables). :

C_1 = x+y+z

C_2 = x^2+y^2+z^2+2X\overline+2Y\overline+2Z\overline

C_3 = xyz - xX\overline - yY\overline - zZ\overline + XYZ + \overline

Where ''x'', ''y'', ''z'' are real-valued and ''X'', ''Y'', ''Z'' are octonion valued. Another way of writing these invariants is as (combinations of) Tr(''M''), Tr(''M''²) and Tr(''M''³) of the

hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

: :

M = \begin
x & \overline & Y \\
Z & y & \overline \\
\overline & X & z
\end

The set of polynomials defines a 24-dimensional compact surface.

Representations

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the

Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...

. The dimensions of the smallest irreducible representations are : :1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912… The 52-dimensional representation is the

adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...

, and the 26-dimensional one is the trace-free part of the action of F₄ on the exceptional

of dimension 27. There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The

s are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the

in the order such that the double arrow points from the second to the third).

References

* *

John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appli ...

, ''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. * * {{String theory topics , state=collapsed Algebraic groups Lie groups Exceptional Lie algebras