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Inmathematics,E₆ is the name of some closely relatedLie groups, linearalgebraic groups or theirLie algebras${\displaystyle {\mathfrak{e}}_6}$, all of which have dimension 78; the same notation E₆ is used for the correspondingroot lattice, which hasrank 6. The designation E₆ comes from the Cartan–Killing classification of the complexsimple Lie algebras (seeÉlie Cartan § Work). This classifies Lie algebras into four infinite series labeled A_n, B_n, C_n, D_n, andfive exceptional cases labeled E₆,E₇,E₈,F₄, andG₂. The E₆ algebra is thus one of the five exceptional cases.

The fundamental group of the adjoint form of E₆ (as a complex or compact Lie group) is thecyclic groupZ/3Z, and itsouter automorphism group is the cyclic groupZ/2Z. For the simply-connected form, itsfundamental representation is 27-dimensional, and a basis is given by the27 lines on a cubic surface. Thedual representation, which is inequivalent, is also 27-dimensional.

Inparticle physics, E₆ plays a role in somegrand unified theories.

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 ofcomplex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental groupZ/3Z, has maximalcompact 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 compact form (which is usually the one meant if no other information is given), which has fundamental groupZ/3Z and outer automorphism groupZ/2Z.
• The split form, EI (or E₆₍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 E₆₍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 groupZ and trivial outer automorphism group.
• EIV (or E_6(-26)), which has maximal compact subgroup F₄, trivial fundamental group cyclic and outer automorphism group of order 2.

The EIV form of E₆ is the group of collineations (line-preserving transformations) of theoctonionic projective planeOP².^[1] It is also the group of determinant-preserving linear transformations of the exceptionalJordan 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 theisometry group of a 32-dimensionalRiemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E₇ and E₈ are known as theRosenfeld projective planes, and are part of theFreudenthal magic square.

By means of aChevalley 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 ofGalois cohomology (over aperfect fieldk) by the setH¹(k, Aut(E₆)) which, because the Dynkin diagram of E₆ (seebelow) has automorphism groupZ/2Z, maps toH¹(k,Z/2Z) = Hom (Gal(k),Z/2Z) with kernelH¹(k, E_6,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 mentionedabove, but with a subtlety concerning the fundamental group: all adjoint forms of E₆ have fundamental groupZ/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 formsEI = E₆₍₆₎ andEIV = E_6(-26) are said to beinner or of type¹E₆ meaning that their class lies inH¹(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 beouter or of type²E₆.

Over finite fields, theLang–Steinberg theorem implies thatH¹(k, E₆) = 0, meaning that E₆ has exactly one twisted form, known as²E₆: seebelow.

Similar to how the algebraic group G₂ is the automorphism group of theoctonions and the algebraic group F₄ is the automorphism group of anAlbert algebra, an exceptionalJordan algebra, the algebraic group E₆ is the group of linear automorphisms of an Albert algebra that preserve a certain cubic form, called the "determinant".^[3]

TheDynkin diagram for E₆ is given by, which may also be drawn as.

Although theyspan a six-dimensional space, it is much more symmetrical to consider them asvectors 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${\displaystyle (\mathbf{3};\mathbf{3};\mathbf{3})}$ where${\displaystyle \mathbf{3}}$ is one of${\displaystyle \left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right),\left(-\frac{1}{3},\frac{2}{3},-\frac{1}{3}\right),\left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right),}$plus all 27 combinations of${\displaystyle (\overline{\mathbf{3}};\overline{\mathbf{3}};\overline{\mathbf{3}})}$ where${\displaystyle \overline{\mathbf{3}}}$ is one of${\displaystyle \left(-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right),\left(\frac{1}{3},-\frac{2}{3},\frac{1}{3}\right),\left(\frac{1}{3},\frac{1}{3},-\frac{2}{3}\right).}$

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)

${\displaystyle \left(\frac{1}{3},-\frac{2}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right)}$

E₆ is the subset of E₈ where a consistent set of three coordinates are equal (e.g. first or last). This facilitates explicit definitions of E₇ and E₆ as:

E₇ = {α ∈Z⁷ ∪ (Z+⁠1/2⁠)⁷ : Σα_i² +α₁² = 2, Σα_i +α₁ ∈ 2Z},

E₆ = {α ∈Z⁶ ∪ (Z+⁠1/2⁠)⁶ : Σα_i² + 2α₁² = 2, Σα_i + 2α₁ ∈ 2Z}

The following 72 E₆ roots are derived in this manner from the split realeven E₈ roots. 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 asubgroup of E₈, is the following:

All${\displaystyle 4\times \left(\begin{array}{c}5\\ 2\end{array}\right)}$ permutations of

${\displaystyle (\pm 1,\pm 1,0,0,0,0)}$ preserving the zero at the last entry,

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

${\displaystyle \left(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{\sqrt{3}}{2}\right).}$

Thus the 78 generators consist of the following subalgebras:

A 45-dimensional SO(10) subalgebra, including the above${\displaystyle 4\times \left(\begin{array}{c}5\\ 2\end{array}\right)}$ generators plus the fiveCartan generators corresponding to the first five entries.

Two 16-dimensional subalgebras that transform as aWeyl spinor of${\displaystyle \mathrm{spin}(10)}$ and its complex conjugate. These have a non-zero last entry.

1 generator which is their chirality generator, and is the sixthCartan generator.

One choice ofsimple roots for E₆ is given by the rows of the following matrix, indexed in the order:

TheWeyl group of E₆ is of order 51840: it is theautomorphism group of the uniquesimple group of order 25920 (which can be described as any of: PSU₄(2), PSΩ₆⁻(2), PSp₄(3) or PSΩ₅(3)).^[4]

The Lie algebra E₆ has an F₄ 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 78-dimensional adjoint representation, there are two dual27-dimensional "vector" representations.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by theWeyl character formula. The dimensions of the smallest irreducible representations are (sequenceA121737 in theOEIS):

1, 27 (twice),78, 351 (four times),650, 1728 (twice),2430,2925,3003 (twice),5824 (twice), 7371 (twice), 7722 (twice), 17550 (twice), 19305 (four times), 34398 (twice),34749,43758, 46332 (twice), 51975 (twice), 54054 (twice), 61425 (twice),70070,78975 (twice),85293, 100386 (twice),105600, 112320 (twice),146432 (twice),252252 (twice), 314496 (twice), 359424 (four times),371800 (twice), 386100 (twice), 393822 (twice), 412776 (twice),442442 (twice)...

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.

Thefundamental representations have dimensions 27, 351, 2925, 351, 27 and 78 (corresponding to the six nodes in theDynkin diagram in the order chosen for theCartan matrix above, i.e., the nodes are read in the five-node chain first, with the last node being connected to the middle one).

The embeddings of the maximal subgroups of E₆ up to dimension 78 are shown to the right.

TheE₆ polytope is theconvex hull of the roots of E₆. It therefore exists in 6 dimensions; itssymmetry group contains theCoxeter group for E₆ as anindex 2 subgroup.

The groups of typeE₆ over arbitrary fields (in particular finite fields) were introduced by Dickson (1901,1908).

The points over afinite field withq elements of the (split) algebraic group E₆ (seeabove), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finiteChevalley group. This is closely connected to the group writtenE₆(q), however there is ambiguity in this notation, which can stand for several things:

• the finite group consisting of the points overF_q of the simply connected form of E₆ (for clarity, this can be writtenE_6,sc(q) or more rarely${\displaystyle {\tilde{E}}_6(q)}$ and is known as the "universal" Chevalley group of type E₆ overF_q),
• (rarely) the finite group consisting of the points overF_q of the adjoint form of E₆ (for clarity, this can be writtenE_6,ad(q), and is known as the "adjoint" Chevalley group of type E₆ overF_q), or
• the finite group which is the image of the natural map from the former to the latter: this is what will be denoted byE₆(q) in the following, as is most common in texts dealing with finite groups.

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 anyq,E_6,sc(q) is itsSchur cover, andE_6,ad(q) lies in its automorphism group; furthermore, whenq−1 is not divisible by 3, all three coincide, and otherwise (whenq is congruent to 1 mod 3), the Schur multiplier of E₆(q) is 3 and E₆(q) is of index 3 inE_6,ad(q), which explains whyE_6,sc(q) andE_6,ad(q) are often written as 3·E₆(q) and E₆(q)·3. From the algebraic group perspective, it is less common for E₆(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 overF_q unlikeE_6,sc(q) andE_6,ad(q).

Beyond this "split" (or "untwisted") form of E₆, there is also one other form of E₆ over the finite fieldF_q, known as²E₆, 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 ofF_q². Twisting does not change the fact that the algebraic fundamental group of²E_6,ad isZ/3Z, but it does change thoseq for which the covering of²E_6,ad by²E_6,sc is non-trivial on theF_q-points. Precisely:²E_6,sc(q) is a covering of²E₆(q), and²E_6,ad(q) lies in its automorphism group; whenq+1 is not divisible by 3, all three coincide, and otherwise (whenq is congruent to 2 mod 3), the degree of²E_6,sc(q) over²E₆(q) is 3 and²E₆(q) is of index 3 in²E_6,ad(q), which explains why²E_6,sc(q) and²E_6,ad(q) are often written as 3·²E₆(q) and²E₆(q)·3.

Two notational issues should be raised concerning the groups²E₆(q). One is that this is sometimes written²E₆(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 theF_q-points of an algebraic group. Another is that whereas²E_6,sc(q) and²E_6,ad(q) are theF_q-points of an algebraic group, the group in question also depends onq (e.g., the points overF_q² of the same group are the untwistedE_6,sc(q²) andE_6,ad(q²)).

The groupsE₆(q) and²E₆(q) are simple for anyq,^[5][6] and constitute two of the infinite families in theclassification of finite simple groups. Their order is given by the following formula (sequenceA008872 in theOEIS):

${\displaystyle |E_6(q)|=\frac{1}{\mathrm{g}\mathrm{c}\mathrm{d}(3,q-1)}{q}^{36}(q^{12}-1)(q^9-1)(q^8-1)(q^6-1)(q^5-1)(q^2-1)}$

${\displaystyle |{}^2{E}_6(q)|=\frac{1}{\mathrm{g}\mathrm{c}\mathrm{d}(3,q+1)}{q}^{36}(q^{12}-1)(q^9+1)(q^8-1)(q^6-1)(q^5+1)(q^2-1)}$

(sequenceA008916 in theOEIS). The order ofE_6,sc(q) orE_6,ad(q) (both are equal) can be obtained by removing the dividing factorgcd(3,q−1) from the first formula (sequenceA008871 in theOEIS), and the order of²E_6,sc(q) or²E_6,ad(q) (both are equal) can be obtained by removing the dividing factorgcd(3,q+1) from the second (sequenceA008915 in theOEIS).

The Schur multiplier ofE₆(q) is alwaysgcd(3,q−1) (i.e.,E_6,sc(q) is its Schur cover). The Schur multiplier of²E₆(q) isgcd(3,q+1) (i.e.,²E_6,sc(q) is its Schur cover) outside of the exceptional caseq=2 where it is 2²·3 (i.e., there is an additional 2²-fold cover). The outer automorphism group ofE₆(q) is the product of the diagonal automorphism groupZ/gcd(3,q−1)Z (given by the action ofE_6,ad(q)), the groupZ/2Z of diagram automorphisms, and the group of field automorphisms (i.e., cyclic of orderf ifq=p^f wherep is prime). The outer automorphism group of²E₆(q) is the product of the diagonal automorphism groupZ/gcd(3,q+1)Z (given by the action of²E_6,ad(q)) and the group of field automorphisms (i.e., cyclic of orderf ifq=p^f wherep is prime).

N = 8 supergravity in five dimensions, which is adimensional reduction fromeleven-dimensional supergravity, admits anE₆ bosonic global symmetry and anSp(8) bosoniclocal symmetry. The fermions are in representations ofSp(8), the gauge fields are in a representation ofE₆, and the scalars are in a representation of both (Gravitons aresinglets with respect to both). Physical states are in representations of the cosetE₆/Sp(8).

Ingrand unification theories,E₆ appears as a possible gauge group which, after itsbreaking, gives rise to theSU(3) ×
SU(2) × U(1)gauge group of theStandard Model. One way of achieving this is through breaking toSO(10) × U(1). The adjoint78 representation breaks, as explained above, into an adjoint45, spinor16 and16 as well as a singlet of theSO(10) subalgebra. Including theU(1) charge we have

${\displaystyle 78\to {45}_0\oplus {16}_{-3}\oplus {\overline{16}}_3+1_0.}$

Where the subscript denotes theU(1) charge.

Likewise, the fundamental representation27 and its conjugate27 break into a scalar1, a vector10 and a spinor, either16 or16:

${\displaystyle 27\to 1_4\oplus {10}_{-2}\oplus {16}_1,}$

${\displaystyle \overline{27}\to 1_{-4}\oplus {10}_2\oplus {\overline{16}}_{-1}.}$

Thus, one can get the Standard Model's elementary fermions and Higgs boson.

• E_n (Lie algebra)
• ADE classification
• Freudenthal magic square

• Adams, J. Frank (1996),Lectures on exceptional Lie groups, Chicago Lectures in Mathematics,University of Chicago Press,ISBN 978-0-226-00526-3,MR 1428422.
• Baez, John (2002)."The Octonions, Section 4.4: E₆".Bull. Amer. Math. Soc.39 (2):145–205.arXiv:math/0105155.doi:10.1090/S0273-0979-01-00934-X.ISSN 0273-0979.S2CID 586512. Online HTML version at[1].
• Cremmer, E.; J. Scherk; J. H. Schwarz (1979). "Spontaneously Broken N=8 Supergravity".Phys. Lett. B.84 (1):83–86.Bibcode:1979PhLB...84...83C.doi:10.1016/0370-2693(79)90654-3.
• Dickson, Leonard Eugene (1901),"A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface",The Quarterly Journal of Pure and Applied Mathematics,33:145–173, reprinted in volume V of his collected works
• Dickson, Leonard Eugene (1908),"A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface (second paper)",The Quarterly Journal of Pure and Applied Mathematics,39:205–209, reprinted in volume VI of his collected works, Chelsea Publishing Company, 1983,ISBN 9780828403061
• Ichiro, Yokota (2009). "Exceptional Lie groups".arXiv:0902.0431 [math.DG].

• Algebraic groups
• Lie groups
• Exceptional Lie algebras

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