E6 (mathematics)

, 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 (see Élie Cartan § Work).

For the simply-connected form, its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface.

In particle physics, E6 plays a role in some grand unified theories.

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 (see below) 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 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 EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane OP2.

[1] 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.

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 (sometimes also known as "untwisted") 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 (over a perfect field k) by the set H1(k, Aut(E6)) which, because the Dynkin diagram of E6 (see below) has automorphism group Z/2Z, maps to H1(k, Z/2Z) = Hom (Gal(k), Z/2Z) with kernel H1(k, E6,ad).

[2] 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 (which may be trivial on the real points); 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(6) and EIV=E6(-26) are said to be inner or of type 1E6 meaning that their class lies in H1(k, E6,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 2E6.

Over finite fields, the Lang–Steinberg theorem implies that H1(k, E6) = 0, meaning that E6 has exactly one twisted form, known as 2E6: see below.

Notice the last 3 dimensions being the same as required: An alternative (6-dimensional) description of the root system, which is useful in considering E6 × SU(3) 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 : The Weyl group of E6 is of order 51840: it is the automorphism group of the unique simple group of order 25920 (which can be described as any of: PSU4(2), PSΩ6−(2), PSp4(3) or PSΩ5(3)).

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 A121737 in the OEIS): The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E6 (equivalently, those whose weights belong to the root lattice 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 points over a finite field with q elements of the (split) algebraic group E6 (see above), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group.

This is closely connected to the group written E6(q), 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(n,q), PGL(n,q) and PSL(n,q), can be summarized as follows: E6(q) is simple for any q, E6,sc(q) is its Schur cover, and E6,ad(q) lies in its automorphism group; furthermore, when q−1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 1 mod 3), the Schur multiplier of E6(q) is 3 and E6(q) is of index 3 in E6,ad(q), which explains why E6,sc(q) and E6,ad(q) are often written as 3·E6(q) and E6(q)·3.

Beyond this "split" (or "untwisted") 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(q), which is known as a Steinberg group, can be seen as the subgroup of E6(q2) 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.

One is that this is sometimes written 2E6(q2), 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.

The order of E6,sc(q) or E6,ad(q) (both are equal) can be obtained by removing the dividing factor gcd(3,q−1) from the first formula (sequence A008871 in the OEIS), and the order of 2E6,sc(q) or 2E6,ad(q) (both are equal) can be obtained by removing the dividing factor gcd(3,q+1) from the second (sequence A008915 in the OEIS).

N = 8 supergravity in five dimensions, which is a dimensional reduction from eleven-dimensional supergravity, admits an E6 bosonic global symmetry and an Sp(8) bosonic local symmetry.

In grand unification theories, E6 appears as a possible gauge group which, after its breaking, gives rise to the SU(3) × SU(2) × U(1) gauge group of the standard model.

Likewise, the fundamental representation 27 and its conjugate 27 break into a scalar 1, a vector 10 and a spinor, either 16 or 16: Thus, one can get the Standard Model's elementary fermions and Higgs boson.