F4 (mathematics)

Last updated September 28, 2024

In mathematics, F₄ is 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.

Algebra

Dynkin diagram

The Dynkin diagram for F₄ is: .

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. The group has ID (1152,157478) in the small groups library.

Cartan matrix

\left[{\begin{array}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{array}}\right]

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 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).

Simple roots

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

{\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}

The Hasse diagram for the F₄ root poset is shown below right.

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

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).

C_{1}=x+y+z

C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}

C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}

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:

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

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).

Embeddings of the maximal subgroups of F₄ up to dimension 273 with associated projection matrix are shown below.

Related Research Articles

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 $O$ or blackboard bold $. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.$

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $under the operation of composition.$

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

<span class="mw-page-title-main">Root system</span> Geometric arrangements of points, foundational to Lie theory

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.

<span class="mw-page-title-main">Dynkin diagram</span> Pictorial 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.

<span class="mw-page-title-main">Simple Lie group</span> Connected non-abelian Lie group lacking nontrivial connected normal subgroups

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 symmetric spaces.

In mathematics, G₂ is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras $as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G 2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.$

In mathematics, E₆ is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras $, all of which have dimension 78; the same notation E 6 is used for the corresponding root lattice, which has rank 6. The designation E 6 comes from the Cartan-Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled A n, B n, C n, D n, and five exceptional cases labeled E 6, E 7, E 8, F 4, and G 2 . The E 6 algebra is thus one of the five exceptional cases.$

In mathematics, the Heisenberg group $, named after Werner Heisenberg, is the group of 3\times3 upper triangular matrices of the form$

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 A_n, B_n, C_n, D_n, and five exceptional cases labeled G₂, F₄, E₆, E₇, and E₈. The E₈ algebra is the largest and most complicated of these exceptional cases.

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 A_n, B_n, C_n, D_n, and five exceptional cases labeled E₆, E₇, E₈, F₄, and G₂. The E₇ algebra is thus one of the five exceptional cases.

<span class="mw-page-title-main">Triality</span> Relationship between certain vector spaces

In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D₄ and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry.

<span class="mw-page-title-main">SO(8)</span> Rotation group in 8-dimensional Euclidean space

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.

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.

<span class="mw-page-title-main">Exceptional object</span>

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others.

<span class="mw-page-title-main">Classical group</span>

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

In mathematics, the Freudenthal magic square is a construction relating several Lie algebras. It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at the right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction.

<span class="mw-page-title-main">Glossary of Lie groups and Lie algebras</span>

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

References

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

Adams, J. Frank (1996). Lectures on exceptional Lie groups. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0-226-00526-3. MR 1428422.
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.
Chevalley C, Schafer RD (February 1950). "The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl. Acad. Sci. U.S.A. 36 (2): 137–41. Bibcode:1950PNAS...36..137C. doi: 10.1073/pnas.36.2.137 . PMC 1063148 . PMID 16588959.
Jacobson, Nathan (1971-06-01). Exceptional Lie Algebras (1st ed.). CRC Press. ISBN 0-8247-1326-5.

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

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F4 (mathematics)

Contents

Algebra

Dynkin diagram

Weyl/Coxeter group

Cartan matrix

F₄ lattice

Roots of F₄

Simple roots

F₄ polynomial invariant

Representations

See also

Related Research Articles

References

F4 (mathematics)

Contents

Algebra

Dynkin diagram

Weyl/Coxeter group

Cartan matrix

F4 lattice

Roots of F4

Simple roots

F4 polynomial invariant

Representations

See also

Related Research Articles

References

F₄ lattice

Roots of F₄

F₄ polynomial invariant