Algebraic structure → Group theoryGroup theory |
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Lie groups and Lie algebras |
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In mathematics, **F _{4}** is a Lie group and also its Lie algebra

- Algebra
- Dynkin diagram
- Weyl/Coxeter group
- Cartan matrix
- F4 lattice
- Roots of F4
- F4 polynomial invariant
- Representations
- See also
- References

The compact real form of F_{4} is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane **OP**^{2}. 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 F_{4} 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 E_{8}.

In older books and papers, F_{4} is sometimes denoted by E_{4}.

The Dynkin diagram for F_{4} is: .

Its Weyl/Coxeter group *G* = *W*(F_{4}) 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 F_{4} 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 F_{4} 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 F_{4}, , is given by the rows of the following matrix:

The Hasse diagram for the F_{4} root poset is shown below right.

Just as O(*n*) is the group of automorphisms which keep the quadratic polynomials *x*^{2} + *y*^{2} + ... invariant, F_{4} 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(*M*^{2}) and Tr(*M*^{3}) of the hermitian octonion matrix:

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

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 F_{4} 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 F_{4} up to dimension 273 with associated projection matrix are shown below.

In mathematics, a **Lie algebra** is a vector space together with an operation called the **Lie bracket**, an alternating bilinear map , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, with the Lie bracket defined as the commutator .

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.

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.

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.

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, **G _{2}** 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

In mathematics, **E _{6}** 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

In mathematics, the **Heisenberg group**, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

In mathematics, **E _{8}** 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 E

In mathematics, **E _{7}** is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras

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 D_{4} 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.

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*(2*n*). **Simple algebraic groups** and (more generally) **semisimple algebraic groups** are reductive.

In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations. The second means that there will be irreducible representations in dimensions greater than 1.

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.

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.

- 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: F_{4}, 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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