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# F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. [1]

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.

## Albert algebra

In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra.

## Cayley plane

In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.

## Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).

## Cubic crystal system

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube.

## Dynkin diagram

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 (drawn as a double or triple line).

## E8 (mathematics)

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.

## Freudenthal magic square

In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).

## Fundamental representation

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

## Hans Freudenthal

Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician.

## Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

## Hurwitz quaternion

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded).

## Hypercubic honeycomb

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn (or B~n-1) for n>.

## 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.

## Jacques Tits

Jacques Tits (born 12 August 1930 in Uccle) is a Belgium-born French mathematician who works on group theory and incidence geometry, and who introduced Tits buildings, the Tits alternative, and the Tits group.

## John C. 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.

## Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## List of simple Lie groups

In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

## 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.

## Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

## Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

## 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.

## Spinor

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

## Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

## Trivial group

In mathematics, a trivial group is a group consisting of a single element.

## University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

## 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.

## Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.

## 24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

## References

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