In mathematics, E₆ is the name of some Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E₆ has rank 6 and dimension 78. The fundamental group of the compact form is the cyclic group Z₃ and its outer automorphism group is the cyclic group Z₂. Its fundamental representation is 27-dimensional (complex). The dual representation, which is inequivalent, is also 27-dimensional. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, a simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, a fundamental representation is a representation of a mathematical structure, such as a group, that satisfies the following condition: All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself. ... If G is a group and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: is the transpose of ρ(g-1) for all g in G. is also a representation, as you may check explicitly. ...

A certain noncompact real form of E₆ is the group of collineations (line-preserving transformations) of the octonionic projective plane OP². It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E₆ has a 27-dimensional complex representation. The compact real form of E₆ is the isometry group of a 32-dimensional Riemannian manifold known as the 'bioctonionic projective plane'. Altogether there are 5 real forms and one complex form. In mathematics, the octonions are a nonassociative extension of the quaternions. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the simple Lie groups were classified by Élie Cartan. ...

In particle physics, E₆ plays a role in some grand unified theories. Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Grand unification, grand unified theory, or GUT is a theory in physics that unifies the strong interaction and electroweak interaction. ...

Algebra

Dynkin diagram

See also Simple Lie group. ... Image File history File links Dynkin_diagram_E6. ...

Roots of E₆

Although they span a six-dimensional space, it's much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space. In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...

(1,−1,0;0,0,0;0,0,0), (−1,1,0;0,0,0;0,0,0),

(−1,0,1;0,0,0;0,0,0), (1,0,−1;0,0,0;0,0,0),

(0,1,−1;0,0,0;0,0,0), (0,−1,1;0,0,0;0,0,0),

(0,0,0;1,−1,0;0,0,0), (0,0,0;−1,1,0;0,0,0),

(0,0,0;−1,0,1;0,0,0), (0,0,0;1,0,−1;0,0,0),

(0,0,0;0,1,−1;0,0,0), (0,0,0;0,−1,1;0,0,0),

(0,0,0;0,0,0;1,−1,0), (0,0,0;0,0,0;−1,1,0),

(0,0,0;0,0,0;−1,0,1), (0,0,0;0,0,0;1,0,−1),

(0,0,0;0,0,0;0,1,−1), (0,0,0;0,0,0;0,−1,1),

All 27 combinations of where is one of , ,

All 27 combinations of where is one of , ,

Simple roots

(0,0,0;0,0,0;0,1,−1)

(0,0,0;0,0,0;1,−1,0)

(0,0,0;0,1,−1;0,0,0)

(0,0,0;1,−1,0;0,0,0)

(0,1,−1;0,0,0;0,0,0)

An alternative description

An alternative (6-dimensional) description of the root system, which is useful in considering as a subgroup of E₈, is the following: In mathematics, E8 is the name of a Lie group and also its Lie algebra . ...

All permutations of

preserving the zero at the last entry,

and all of the following roots with an even number of plus signs

.

Thus the generators comprise of a 45-dimensional subalgebra as well as 32 generators that transform as a Majorana spinor of and its chirality generator. In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

The simple roots in this description are See also Simple Lie group. ...

(-1/2,-1/2,-1/2,-1/2,-1/2,)

(1,1,0,0,0,0)

(0,-1,1,0,0,0)

(0,0,-1,1,0,0)

(0,0,0,-1,1,0)

(0,0,0,0,-1,1)

(-1,1,0,0,0,0)

we have ordered them so that their corresponding nodes in the dynkin diagram are ordered from left to right (in the diagram depicted above) with the side node last. See also Simple Lie group. ...

Cartan matrix

In mathematics, the term Cartan matrix has two meanings. ...

Important subalgebras and representations

The Lie algebra E₆ has an F₄ subalgebra, which is the fixed subalgebra of an outer automorphism, and an subalgebra. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of and .

In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional "vector" representations. In mathematics, E8 is the name of a Lie group and also its Lie algebra . ...

E6 polytope

The E₆ polytope is the convex hull of the roots of E₆. It therefore exists in 6 dimensions; its symmetry group contains the Coxeter group for E₆ as an index 2 subgroup. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. (Note that X may be the union of any set of objects made of points). ... The symmetry group of an object (e. ... In mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ...

Importance in physics

N=8 supergravity in five dimensions, which is a dimensional reduction from 11 dimensional supergravity, admits an E₆ bosonic global symmetry and an bosonic local symmetry. The fermions are in representations of , the gauge fields are in a representation of E₆, and the scalars are in a representation of both (Gravitons are singlets with respect to both). Physical states are in representations of the coset E₆ / SP(8). In theoretical physics, a supergravity theory is a field theory combining supersymmetry and general relativity. ... In statistics, dimensionality reduction can be divided into two categories: feature selection and feature extraction. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In theoretical physics, a singlet usually refers to a one-dimensional representation (e. ...

In grand unification theories, E₆ appears as a possible gauge group which, after its breaking, gives rise to the gauge group of the standard model (also see Importance in physics of E8). This article or section does not cite its references or sources. ... Promotional picture Symmetry Breaking is a rock band from Northern New Jersey, in the United States. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In mathematics, E8 is the name of a Lie group and also its Lie algebra . ...

References

• John Baez, The Octonions, Section 4.4: E₆, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at [1].

John Carlos Baez (b. ...

See also

En (Lie algebra) In mathematics, En is the Kac-Moody algebra whose Dynkin diagram is a line of n-1 points with an extra point attached to the third point from the end. ...

• ADE classification