F4 (mathematics)

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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.

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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.

Algebra

Dynkin diagram

The Dynkin diagram for F4 is: Dyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.pngDyn-node.png.

Weyl/Coxeter group

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, [1] which is realized by the action on the 24-cell.

Cartan matrix

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 vertices of a 24-cell centered at the origin.

Roots of F4

The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F4 in this Coxeter plane projection F4 roots by 24-cell duals.svg
The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F4 in this Coxeter plane projection

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:CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png

Dual 24-cell vertices:CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png

Simple roots

Hasse diagram of F4 root poset with edge labels identifying added simple root position F4HassePoset.svg
Hasse diagram of F4 root poset with edge labels identifying added simple root position

One choice of simple roots for F4, Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-4b.pngDyn2-node n3.pngDyn2-3.pngDyn2-node n4.png, is given by the rows of the following matrix:

F4 polynomial invariant

Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ... 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).

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(M2) and Tr(M3) of the hermitian octonion matrix:

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. The dimensions of the smallest irreducible representations are (sequence A121738 in the OEIS ):

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, and the 26-dimensional one is the trace-free part of the action of F4 on the exceptional Albert algebra of dimension 27.

There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The fundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagram in the order such that the double arrow points from the second to the third).

See also

Related Research Articles

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Quaternion group

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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. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

Dynkin diagram Pictoral representation of symmetry

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled. Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram correspond to important features of the associated Lie algebra.

Simple Lie group Connected non-abelian Lie group lacking nontrivial connected normal subgroups

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G<sub>2</sub> (mathematics) Simple Lie group; the automorphism group of the octonions

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E<sub>6</sub> (mathematics) 78-dimensional exceptional simple Lie group

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E<sub>8</sub> (mathematics) 248-dimensional exceptional simple Lie group

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.

E<sub>7</sub> (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases.

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Exceptional object

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Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

References

  1. Saunders, Neil (2014). "Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups". arXiv: 0812.0182 [math.GR].