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Infinite dimensional Lie group

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In mathematics, E_{6} 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 E_{6} is used for the corresponding root lattice, which has rank 6. The designation E_{6} comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled A_{n}, B_{n}, C_{n}, D_{n}, and five exceptional cases labeled E_{6}, E_{7}, E_{8}, F_{4}, and G_{2}. 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 Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27dimensional.
In particle physics, E_{6} plays a role in some grand unified theories.
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 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 E_{6}, and has an outer automorphism group noncyclic 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 nonalgebraic covers, giving further real forms), all of real dimension 78, as follows:
The EIV form of E_{6} is the group of collineations (linepreserving transformations) of the octonionic projective plane OP^{2}.^{[1]} It is also the group of determinantpreserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27dimensional, which explains why the compact real form of E_{6} has a 27dimensional complex representation. The compact real form of E_{6} is the isometry group of a 32dimensional Riemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E_{7} and E_{8} 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 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 socalled 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 (over a perfect field 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}).^{[2]}
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 noncompact real Lie group forms of E_{6} are therefore not algebraic and admit no faithful finitedimensional 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 ^{1}E_{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 ^{2}E_{6}.
Over finite fields, the Lang–Steinberg theorem implies that H^{1}(k, E_{6}) = 0, meaning that E_{6} has exactly one twisted form, known as ^{2}E_{6}: see below.
The Dynkin diagram for E_{6} is given by , which may also be drawn as .
Although they span a sixdimensional space, it is much more symmetrical to consider them as vectors in a sixdimensional subspace of a ninedimensional 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:
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:
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 (6dimensional) 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
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 E_{6} is given by the rows of the following matrix, indexed in the order :
The Weyl group of E_{6} is of order 51840: it is the automorphism group of the unique simple group of order 25920 (which can be described as any of: PSU_{4}(2), PSΩ_{6}^{−}(2), PSp_{4}(3) or PSΩ_{5}(3)).^{[3]}
The Lie algebra E_{6} has an F_{4} subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) × SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2).
In addition to the 78dimensional adjoint representation, there are two dual 27dimensional "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 (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 E_{6} (equivalently, those whose weights belong to the root lattice of E_{6}), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E_{6}.
The symmetry of the Dynkin diagram of E_{6} explains why many dimensions occur twice, the corresponding representations being related by the nontrivial 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 (corresponding to the six nodes in the Dynkin diagram in the order chosen for the Cartan matrix above, i.e., the nodes are read in the fivenode chain first, with the last node being connected to the middle one).
The E_{6} polytope is the convex hull of the roots of E_{6}. It therefore exists in 6 dimensions; its symmetry group contains the Coxeter group for E_{6} as an index 2 subgroup.
The groups of type E_{6} over arbitrary fields (in particular finite fields) were introduced by Dickson (1901, 1908).
The points over a finite field with q elements of the (split) algebraic group E_{6} (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 E_{6}(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: E_{6}(q) is simple for any q, E_{6,sc}(q) is its Schur cover, and E_{6,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 E_{6}(q) is 3 and E_{6}(q) is of index 3 in E_{6,ad}(q), which explains why E_{6,sc}(q) and E_{6,ad}(q) are often written as 3·E_{6}(q) and E_{6}(q)·3. From the algebraic group perspective, it is less common for E_{6}(q) 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 F_{q} unlike E_{6,sc}(q) and E_{6,ad}(q).
Beyond this “split” (or “untwisted”) form of E_{6}, there is also one other form of E_{6} over the finite field F_{q}, known as ^{2}E_{6}, which is obtained by twisting by the nontrivial automorphism of the Dynkin diagram of E_{6}. Concretely, ^{2}E_{6}(q), which is known as a Steinberg group, can be seen as the subgroup of E_{6}(q^{2}) fixed by the composition of the nontrivial diagram automorphism and the nontrivial field automorphism of F_{q2}. Twisting does not change the fact that the algebraic fundamental group of ^{2}E_{6,ad} is Z/3Z, but it does change those q for which the covering of ^{2}E_{6,ad} by ^{2}E_{6,sc} is nontrivial on the F_{q}points. Precisely: ^{2}E_{6,sc}(q) is a covering of ^{2}E_{6}(q), and ^{2}E_{6,ad}(q) lies in its automorphism group; when q+1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 2 mod 3), the degree of ^{2}E_{6,sc}(q) over ^{2}E_{6}(q) is 3 and ^{2}E_{6}(q) is of index 3 in ^{2}E_{6,ad}(q), which explains why ^{2}E_{6,sc}(q) and ^{2}E_{6,ad}(q) are often written as 3·^{2}E_{6}(q) and ^{2}E_{6}(q)·3.
Two notational issues should be raised concerning the groups ^{2}E_{6}(q). One is that this is sometimes written ^{2}E_{6}(q^{2}), 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 F_{q}points of an algebraic group. Another is that whereas ^{2}E_{6,sc}(q) and ^{2}E_{6,ad}(q) are the F_{q}points of an algebraic group, the group in question also depends on q (e.g., the points over F_{q2} of the same group are the untwisted E_{6,sc}(q^{2}) and E_{6,ad}(q^{2})).
The groups E_{6}(q) and ^{2}E_{6}(q) are simple for any q,^{[4]}^{[5]} and constitute two of the infinite families in the classification of finite simple groups. Their order is given by the following formula (sequence A008872 in the OEIS):
(sequence A008916 in the OEIS). The order of E_{6,sc}(q) or E_{6,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 ^{2}E_{6,sc}(q) or ^{2}E_{6,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).
The Schur multiplier of E_{6}(q) is always gcd(3,q−1) (i.e., E_{6,sc}(q) is its Schur cover). The Schur multiplier of ^{2}E_{6}(q) is gcd(3,q+1) (i.e., ^{2}E_{6,sc}(q) is its Schur cover) outside of the exceptional case q=2 where it is 2^{2}·3 (i.e., there is an additional 2^{2}fold cover). The outer automorphism group of E_{6}(q) is the product of the diagonal automorphism group Z/gcd(3,q−1)Z (given by the action of E_{6,ad}(q)), the group Z/2Z of diagram automorphisms, and the group of field automorphisms (i.e., cyclic of order f if q=p^{f} where p is prime). The outer automorphism group of ^{2}E_{6}(q) is the product of the diagonal automorphism group Z/gcd(3,q+1)Z (given by the action of ^{2}E_{6,ad}(q)) and the group of field automorphisms (i.e., cyclic of order f if q=p^{f} where p is prime).
N = 8 supergravity in five dimensions, which is a dimensional reduction from 11 dimensional supergravity, admits an E_{6} bosonic global symmetry and an Sp(8) bosonic local symmetry. The fermions are in representations of Sp(8), the gauge fields are in a representation of E_{6}, and the scalars are in a representation of both (Gravitons are singlets with respect to both). Physical states are in representations of the coset E_{6}/Sp(8).
In grand unification theories, E_{6} 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 (also see Importance in physics of E8). One way of achieving this is through breaking to SO(10) × U(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45, spinor 16 and 16 as well as a singlet of the SO(10) subalgebra. Including the U(1) charge we have
Where the subscript denotes the U(1) charge.
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.