In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, E
6 is the name of some closely related
Lie groups, linear
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
s or their
Lie algebras
, all of which have dimension 78; the same notation E
6 is used for the corresponding
root lattice
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 ...
, which has
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
6. The designation E
6 comes from the Cartan–Killing classification of the complex
simple Lie algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.
A direct sum of s ...
s (see ). This classifies Lie algebras into four infinite series labeled A
''n'', B
''n'', C
''n'', D
''n'', and
five exceptional cases labeled E
6,
E7,
E8,
F4, and
G2. The E
6 algebra is thus one of the five exceptional cases.
The fundamental group of the complex form, compact real form, or any algebraic version of E
6 is the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
Z/3Z, and its
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. The outer automorphism group is usually denoted . If is trivial and has a t ...
is the cyclic group Z/2Z. Its
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group
or Lie algebra whose highest weig ...
is 27-dimensional (complex), and a basis is given by the
27 lines on a cubic surface. The
dual representation
In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:
: is the transpose of , that is, = for all .
The dual representation ...
, which is inequivalent, is also 27-dimensional.
In
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, E
6 plays a role in some
grand unified theories
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this u ...
.
Real and complex forms
There is a unique complex Lie algebra of type E
6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E
6 of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
subgroup the compact form (see below) of E
6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.
As well as the complex Lie group of type E
6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:
* The compact form (which is usually the one meant if no other information is given), which has fundamental group Z/3Z and outer automorphism group Z/2Z.
* The split form, EI (or E
6(6)), which has maximal compact subgroup Sp(4)/(±1), fundamental group of order 2 and outer automorphism group of order 2.
* The quasi-split form EII (or E
6(2)), which has maximal compact subgroup SU(2) × SU(6)/(center), fundamental group cyclic of order 6 and outer automorphism group of order 2.
* EIII (or E
6(-14)), which has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z and trivial outer automorphism group.
* EIV (or E
6(-26)), which has maximal compact subgroup F
4, trivial fundamental group cyclic and outer automorphism group of order 2.
The EIV form of E
6 is the group of collineations (line-preserving transformations) of the
octonionic projective plane
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.Baez (2002).
The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describin ...
OP
2. It is also the group of determinant-preserving linear transformations of the exceptional
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E
6 has a 27-dimensional complex representation. The compact real form of E
6 is the
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. Its identity element is the ...
of a 32-dimensional
Riemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E
7 and E
8 are known as the
Rosenfeld projective plane
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea indep ...
s, and are part of the
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). It is named after Hans Freudenthal and Jacques Tits, who developed the idea i ...
.
E6 as an algebraic group
By means of a
Chevalley basis In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite f ...
for the Lie algebra, one can define E
6 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") adjoint form of E
6. Over an algebraically closed field, this and its triple cover are the only forms; however, over other fields, there are often many other forms, or "twists" of E
6, which are classified in the general framework of
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
(over a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k' ...
''k'') by the set ''H''
1(''k'', Aut(E
6)) which, because the Dynkin diagram of E
6 (see
below) has automorphism group Z/2Z, maps to ''H''
1(''k'', Z/2Z) = Hom (Gal(''k''), Z/2Z) with kernel ''H''
1(''k'', E
6,ad).
Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E
6 coincide with the three real Lie groups mentioned
above, but with a subtlety concerning the fundamental group: all adjoint forms of E
6 have fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); the further non-compact real Lie group forms of E
6 are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E
6 as well as the noncompact forms EI=E
6(6) and EIV=E
6(-26) are said to be ''inner'' or of type
1E
6 meaning that their class lies in ''H''
1(''k'', E
6,ad) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be ''outer'' or of type
2E
6.
Over finite fields, the
Lang–Steinberg theorem In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if ''G'' is a connected smooth algebraic group over a finite field \mathbf_q, then, writing \sigma: G \to G, \, x \mapsto x^q for the Frobenius, the morphism of varieties
:G ...
implies that ''H''
1(''k'', E
6) = 0, meaning that E
6 has exactly one twisted form, known as
2E
6: see
below.
Automorphisms of an Albert Algebra
Similar to how the algebraic group G
2 is the automorphism group of the
octonion
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 or blackboard bold \mathbb O. Octonions hav ...
s and the algebraic group F
4 is the automorphism group of an
Albert algebra, an exceptional
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
, the algebraic group E
6 is the group of linear automorphisms of an Albert algebra that preserve a certain cubic form, called the "determinant".
Algebra
Dynkin diagram
The
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
for E
6 is given by , which may also be drawn as .
Roots of E6
Although they
span
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan ester ...
a six-dimensional space, it is much more symmetrical to consider them as
vectors in a six-dimensional subspace of a nine-dimensional space. Then one can take the roots to be
:(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),
plus all 27 combinations of
where
is one of
plus all 27 combinations of
where
is one of
Simple roots
One possible selection for the simple roots of E6 is:
:(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)
:
E6 roots derived from the roots of E8
E
6 is the subset of E
8 where a consistent set of three coordinates are equal (e.g. first or last). This facilitates explicit definitions of E
7 and E
6 as:
:E
''7'' = ,
:E
''6'' =
The following 72 E6 roots are derived in this manner from the split real
even E8 roots. Notice the last 3 dimensions being the same as required:
:
An alternative description
An alternative (6-dimensional) description of the root system, which is useful in considering E
6 × SU(3) as a
subgroup of E
8, is the following:
All
permutations of
:
preserving the zero at the last entry,
and all of the following roots with an odd number of plus signs
:
Thus the 78 generators consist of the following subalgebras:
: A 45-dimensional SO(10) subalgebra, including the above
generators plus the five
Cartan generators corresponding to the first five entries.
: Two 16-dimensional subalgebras that transform as a
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
of
and its complex conjugate. These have a non-zero last entry.
: 1 generator which is their chirality generator, and is the sixth
Cartan generator.
One choice of
simple root
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
s for E
6 is given by the rows of the following matrix, indexed in the order
:
: