E6 (mathematics)


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 E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras. This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.
The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional.
In particle physics, E6 plays a role in some grand unified theories.

Real and complex forms

There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form of E6, 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 E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center, all of real dimension 78, as follows:
The EIV form of E6 is the group of collineations 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'; similar constructions for E7 and E8 are known as the Rosenfeld projective planes, and are part of the Freudenthal magic square.

E6 as an algebraic group

By means of a Chevalley basis for the Lie algebra, one can define E6 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 adjoint form of E6. 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 E6, which are classified in the general framework of Galois cohomology by the set H1 which, because the Dynkin diagram of E6 has automorphism group Z/2Z, maps to H1 = Hom with kernel H1.
Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E6 coincide with the three real Lie groups mentioned [|above], but with a subtlety concerning the fundamental group: all adjoint forms of E6 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 ; the further non-compact real Lie group forms of E6 are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E6 as well as the noncompact forms EI=E6 and EIV=E6 are said to be inner or of type 1E6 meaning that their class lies in H1 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 2E6.
Over finite fields, the Lang–Steinberg theorem implies that H1 = 0, meaning that E6 has exactly one twisted form, known as 2E6: see [|below].

Automorphisms of an Albert Algebra

Similar to how the algebraic group G2 is the automorphism group of the octonions and the algebraic group F2 is the automorphism group of an Albert algebra, an exceptional Jordan algebra, the algebraic group E6 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 for E6 is given by, which may also be drawn as.

Roots of E6

Although they span 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
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:
of E6 root poset with edge labels identifying added simple root position

E6 roots derived from the roots of E8

E6 is the subset of E8 where a consistent set of three coordinates are equal. This facilitates explicit definitions of E7 and E6 as:
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 description of the root system, which is useful in considering E6 × SU as a subgroup of E8, is the following:
All permutations of
and all of the following roots with an odd number of plus signs
Thus the 78 generators consist of the following subalgebras:
One choice of simple roots for E6 is given by the rows of the following matrix, indexed in the order :

Weyl group

The Weyl group of E6 is of order 51840: it is the automorphism group of the unique simple group of order 25920, PSΩ6−, PSp4 or PSΩ5).

Cartan matrix

Important subalgebras and representations

The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU × SU × SU subalgebra. Other maximal subalgebras which have an importance in physics and can be read off the Dynkin diagram, are the algebras of SO × U and SU × SU.
In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional "vector" 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 :
The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E6, whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E6.
The symmetry of the Dynkin diagram of E6 explains why many dimensions occur twice, the corresponding representations being related by the non-trivial outer automorphism; however, there are sometimes even more representations than this, such as four of dimension 351, two of which are fundamental and two of which are not.
The fundamental representations have dimensions 27, 351, 2925, 351, 27 and 78.

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.

Chevalley and Steinberg groups of type E6 and 2E6

The groups of type E6 over arbitrary fields were introduced by.
The points over a finite field with q elements of the algebraic group E6, whether of the adjoint or simply connected form, give a finite Chevalley group. This is closely connected to the group written E6, however there is ambiguity in this notation, which can stand for several things:
From the finite group perspective, the relation between these three groups, which is quite analogous to that between SL, PGL and PSL, can be summarized as follows: E6 is simple for any q, E6,sc is its Schur cover, and E6,ad lies in its automorphism group; furthermore, when q−1 is not divisible by 3, all three coincide, and otherwise, the Schur multiplier of E6 is 3 and E6 is of index 3 in E6,ad, which explains why E6,sc and E6,ad are often written as 3·E6 and E6·3. From the algebraic group perspective, it is less common for E6 to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraic group over Fq unlike E6,sc and E6,ad.
Beyond this “split” form of E6, there is also one other form of E6 over the finite field Fq, known as 2E6, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E6. Concretely, 2E6, which is known as a Steinberg group, can be seen as the subgroup of E6 fixed by the composition of the non-trivial diagram automorphism and the non-trivial field automorphism of Fq2. Twisting does not change the fact that the algebraic fundamental group of 2E6,ad is Z/3Z, but it does change those q for which the covering of 2E6,ad by 2E6,sc is non-trivial on the Fq-points. Precisely: 2E6,sc is a covering of 2E6, and 2E6,ad lies in its automorphism group; when q+1 is not divisible by 3, all three coincide, and otherwise, the degree of 2E6,sc over 2E6 is 3 and 2E6 is of index 3 in 2E6,ad, which explains why 2E6,sc and 2E6,ad are often written as 3·2E6 and 2E6·3.
Two notational issues should be raised concerning the groups 2E6. One is that this is sometimes written 2E6, a notation which has the advantage of transposing more easily to the Suzuki and Ree groups, but the disadvantage of deviating from the notation for the Fq-points of an algebraic group. Another is that whereas 2E6,sc and 2E6,ad are the Fq-points of an algebraic group, the group in question also depends on q and E6,ad).
The groups E6 and 2E6 are simple for any q, and constitute two of the infinite families in the classification of finite simple groups. Their order is given by the following formula :
. The order of E6,sc or E6,ad can be obtained by removing the dividing factor gcd from the first formula, and the order of 2E6,sc or 2E6,ad can be obtained by removing the dividing factor gcd from the second.
The Schur multiplier of E6 is always gcd. The Schur multiplier of 2E6 is gcd outside of the exceptional case q=2 where it is 22·3. The outer automorphism group of E6 is the product of the diagonal automorphism group Z/gcdZ, the group Z/2Z of diagram automorphisms, and the group of field automorphisms. The outer automorphism group of 2E6 is the product of the diagonal automorphism group Z/gcdZ and the group of field automorphisms.

Importance in physics

in five dimensions, which is a dimensional reduction from dimensional supergravity, admits an bosonic global symmetry and an bosonic local symmetry. The fermions are in representations of, the gauge fields are in a representation of, and the scalars are in a representation of both. Physical states are in representations of the coset.
In grand unification theories, appears as a possible gauge group which, after its breaking, gives rise to gauge group of the standard model. One way of achieving this is through breaking to. The adjoint representation breaks, as explained above, into an adjoint, spinor and as well as a singlet of the subalgebra. Including the charge we have
Where the subscript denotes the charge.
Likewise, the fundamental representation and its conjugate break into a scalar, a vector and a spinor, either or :
Thus, one can get the Standard Model's elementary fermions and Higgs boson.