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{{Short description|248-dimensional exceptional simple Lie group}}
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In [[mathematics]], '''E<sub>8</sub>''' is any of several closely related [[exceptional simple Lie group]]s, linear [[algebraic group]]s or Lie algebras of [[dimension]] 248; the same notation is used for the corresponding [[root lattice]], which has [[Rank of a Lie group|rank]]&nbsp;8. The designation E<sub>8</sub> comes from the [[Killing form|Cartan–Killing classification]] of the complex [[simple Lie algebra]]s, which fall into four infinite series labeled A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, D<sub>''n''</sub>, and [[Exceptional simple Lie group|five exceptional cases]] labeled [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and E<sub>8</sub>. The E<sub>8</sub> algebra is the largest and most complicated of these exceptional cases.

== Basic description ==
The [[Lie group]] E<sub>8</sub> has dimension 248. Its [[Cartan subgroup|rank]], which is the dimension of its [[maximal torus]], is eight.

Therefore, the vectors of the root system are in eight-dimensional [[Euclidean space]]: they are described explicitly later in this article. The [[Weyl group]] of E<sub>8</sub>, which is the [[symmetry group|group of symmetries]] of the maximal torus that are induced by [[conjugacy class|conjugations]] in the whole group, has order 2{{sup|14}}{{nnbsp}}3{{sup|5}}{{nnbsp}}5{{sup|2}}{{nnbsp}}7 = {{gaps|696|729|600}}.

The compact group E<sub>8</sub> is unique among simple compact Lie groups in that its non-[[trivial (mathematics)|trivial]] representation of smallest dimension is the [[Adjoint representation of a Lie algebra|adjoint representation]] (of dimension 248) acting on the Lie algebra E<sub>8</sub> itself; it is also the unique one that has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length).

There is a Lie algebra [[En (Lie algebra)|E<sub>''k''</sub>]] for every integer ''k''&nbsp;≥&nbsp;3. The largest value of ''k'' for which E<sub>''k''</sub> is finite-dimensional is ''k''&nbsp;=&nbsp;8, that is, E<sub>''k''</sub> is infinite-dimensional for any  ''k''&nbsp;>&nbsp;8.

== Real and complex forms ==
There is a unique complex Lie algebra of type E<sub>8</sub>, corresponding to a complex group of complex dimension 248. The complex Lie group E<sub>8</sub> of [[complex dimension]] 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal [[Compact space|compact]] subgroup the compact form (see below) of E<sub>8</sub>, and has an outer automorphism group of order 2 generated by complex conjugation.

As well as the complex Lie group of type E<sub>8</sub>, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:

* The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
* The split form, EVIII (or E<sub>8(8)</sub>), which has maximal compact subgroup Spin(16)/('''Z'''/2'''Z'''), fundamental group of order 2 (implying that it has a [[Double covering group|double cover]], which is a simply connected Lie real group but is not algebraic, see [[#E8 as an algebraic group|below]]) and has trivial outer automorphism group.
* EIX (or E<sub>8(−24)</sub>), which has maximal compact subgroup E<sub>7</sub>&nbsp;&times;&nbsp;SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.

For a complete list of real forms of simple Lie algebras, see the [[list of simple Lie groups]].

== E<sub>8</sub> as an algebraic group ==
By means of a [[Chevalley basis]] for the Lie algebra, one can define E<sub>8</sub> 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") form of E<sub>8</sub>. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or "twists" of E<sub>8</sub>, which are classified in the general framework of [[Galois cohomology]] (over a [[perfect field]] ''k'') by the set H<sup>1</sup>(''k'',Aut(E<sub>8</sub>)), which, because the Dynkin diagram of E<sub>8</sub> (see [[#Dynkin diagram|below]]) has no automorphisms, coincides with H<sup>1</sup>(''k'',E<sub>8</sub>).<ref>{{citation |last1=Платонов |first1=Владимир П. |last2=Рапинчук |first2=Андрей С. |title=Алгебраические группы и теория чисел |year=1991 |publisher=Наука |isbn=5-02-014191-7}} (English translation: {{citation |last1=Platonov |first1=Vladimir P. |last2=Rapinchuk |first2=Andrei S. |title=Algebraic groups and number theory |year=1994 |publisher=Academic Press |isbn=0-12-558180-7}}), §2.2.4</ref>

Over '''R''', the real connected component of the identity of these algebraically twisted forms of E<sub>8</sub> coincide with the three real Lie groups mentioned [[#Real and complex forms|above]], but with a subtlety concerning the fundamental group: all forms of E<sub>8</sub> are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E<sub>8</sub> are therefore not algebraic and admit no faithful finite-dimensional representations.

Over finite fields, the [[Lang–Steinberg theorem]] implies that H<sup>1</sup>(''k'',E<sub>8</sub>)&nbsp;=&nbsp;0, meaning that E<sub>8</sub> has no twisted forms: see [[#Chevalley groups of type E8|below]].

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 {{OEIS|id=A121732}}:
: 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960...

The 248-dimensional representation is the [[Adjoint representation of a Lie group|adjoint representation]].  There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 {{OEIS|id=A181746}}).  The [[fundamental representation]]s are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order chosen for the [[#Cartan matrix|Cartan matrix]] below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).

The coefficients of the character formulas for infinite dimensional irreducible [[representation theory|representation]]s of E<sub>8</sub> depend on some large square matrices consisting of polynomials, the [[Lusztig–Vogan polynomial]]s, an analogue of [[Kazhdan–Lusztig polynomial]]s introduced for [[reductive group]]s in general by [[George Lusztig]] and [[David Kazhdan]] (1983). The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.

These matrices were computed after four years of collaboration by a [[atlas of Lie groups and representations|group of 18 mathematicians and computer scientists]], led by [[Jeffrey Adams (mathematician)|Jeffrey Adams]], with much of the programming done by [[Fokko du Cloux]]. The most difficult case (for exceptional groups)  is the split [[real form]] of  E<sub>8</sub> (see above), where the largest matrix is  of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups  have been known for some time; the calculation for the split form of ''E''<sub>8</sub> is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.

The representations of the E<sub>8</sub> groups over finite fields are given by [[Deligne–Lusztig theory]].

== Constructions ==
One can construct the (compact form of the) E<sub>8</sub> group as the [[automorphism group]] of the corresponding '''e'''<sub>8</sub> Lie algebra. This algebra has a 120-dimensional subalgebra '''so'''(16) generated by ''J''<sub>''ij''</sub> as well as 128 new generators ''Q''<sub>''a''</sub> that transform as a [[Weyl–Majorana spinor]] of '''spin'''(16). These statements determine the commutators
: <math>\left[J_{ij}, J_{k\ell}\right] = \delta_{jk}J_{i\ell} - \delta_{j\ell}J_{ik} - \delta_{ik}J_{j\ell} + \delta_{i\ell}J_{jk}</math>
as well as
: <math>\left[J_{ij}, Q_a\right] = \frac 14 \left(\gamma_i\gamma_j-\gamma_j\gamma_i\right)_{ab} Q_b,</math>
while the remaining commutators (not anticommutators!) between the spinor generators are defined as
: <math>\left[Q_a, Q_b\right] = \gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.</math>

It is then possible to check that the [[Jacobi identity]] is satisfied.

== Geometry ==
The compact real form of E<sub>8</sub> is the [[isometry group]] of the 128-dimensional exceptional compact [[Riemannian symmetric space]] EVIII (in Cartan's [[Riemannian symmetric space#Classification of Riemannian symmetric spaces|classification]]). It is known informally as the "[[octooctonionic projective plane]]" because it can be built using an algebra that is the tensor product of the [[octonion]]s with themselves, and is also known as a [[Rosenfeld projective plane]], though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the [[Freudenthal magic square|''magic square'']], due to [[Hans Freudenthal]] and [[Jacques Tits]] {{harv|Landsberg|Manivel|2001}}.

== E<sub>8</sub> root system ==
[[File:e8 roots zome.jpg|thumb|[[Zome]] model of the E<sub>8</sub> root system, projected into three-space, and represented by the vertices of the [[4 21 polytope|4<sub>21</sub> polytope]], {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}]]
[[File:4 21 E8 to 3D H3 symmetry concentric hulls.png|thumb|Shown in 3D projection using the basis vectors [''u'',''v'',''w''] giving H3 symmetry:
{{ubl
 | ''u'' {{=}} (1, ''φ'', 0, −1, ''φ'', 0,0,0)
 | ''v'' {{=}} (''φ'', 0, 1, ''φ'', 0, −1,0,0)
 | ''w'' {{=}} (0, 1, ''φ'', 0, −1, ''φ'',0,0)
}}

The projected [[4 21 polytope|4<sub>21</sub> polytope]] vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show:
{{ordered list
 | list_style = list-style-position:inside; margin:0;
 | 4 points at the origin
 | 2 icosahedrons
 | 2 dodecahedrons
 | 4 icosahedrons
 | 1 icosidodecahedron
 | 2 dodecahedrons
 | 2 icosahedrons
 | 1 icosidodecahedron
}}

for 240 vertices. These are two concentric sets of hulls from the H4 symmetry of the [[600-cell]] scaled by the golden ratio.<ref>{{Cite web|url=https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/|title=The 600-Cell (Part 1)|date=December 16, 2017}}</ref>
]]

A [[root system]] of rank ''r'' is a particular finite configuration of vectors, called ''roots'', which span an ''r''-dimensional [[Euclidean space]] and satisfy certain geometrical properties. In particular, the root system must be invariant under [[Reflection (mathematics)|reflection]] through the hyperplane perpendicular to any root.

The '''E<sub>8</sub> root system''' is a rank 8 root system containing 240 root vectors spanning '''R'''<sup>8</sup>. It is [[Irreducibility (mathematics)|irreducible]] in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E<sub>8</sub> have the same length. It is convenient for a number of purposes to normalize them to have length {{radic|2}}.  These 240 vectors are the vertices of a [[semi-regular polytope]] discovered by [[Thorold Gosset]] in 1900, sometimes known as the [[4 21 polytope|4<sub>21</sub> polytope]].

=== Construction ===
In the so-called ''even coordinate system'', E<sub>8</sub> is given as the set of all vectors in '''R'''<sup>8</sup> with length squared equal to 2 such that coordinates are either all [[integer]]s or all [[half-integer]]s and the sum of the coordinates is even.

Explicitly, there are 112 roots with integer entries obtained from
: <math>\left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,</math>

by taking an arbitrary combination of signs and an arbitrary [[permutation]] of coordinates, and 128 roots with half-integer entries obtained from
: <math>\left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \,</math>

by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.

[[File:E8-with-thread.jpg|thumb|E<sub>8</sub> 2d projection with thread made by hand]]

The 112 roots with integer entries form a D<sub>8</sub> root system. The E<sub>8</sub> root system also contains a copy of A<sub>8</sub> (which has 72 roots) as well as [[E6 (mathematics)|E<sub>6</sub>]] and [[E7 (mathematics)|E<sub>7</sub>]] (in fact, the latter two are usually ''defined'' as subsets of E<sub>8</sub>).

In the ''odd coordinate system'', E<sub>8</sub> is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.

=== Dynkin diagram ===
The [[Dynkin diagram]] for E<sub>8</sub> is given by [[File:Dynkin diagram E8.svg|120px]].

This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots that are not joined by a line are [[orthogonal]].

=== Cartan matrix ===
The [[Cartan matrix]] of a rank {{math|''r''}} root system is an {{math|''r × r''}} [[Matrix (mathematics)|matrix]] whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
: <math>A_{ij} = 2\frac{\left(\alpha_i, \alpha_j\right)}{\left(\alpha_i, \alpha_i\right)}</math>
where {{math|(&nbsp;,&nbsp;)}} is the Euclidean [[inner product]] and {{math|''α<sub>i</sub>''}} are the simple roots. The entries are independent of the choice of simple roots (up to ordering).

The Cartan matrix for E<sub>8</sub> is given by
: <math>\left [
\begin{array}{rr}
 2 & -1 &  0 &  0 &  0 &  0 &  0 & 0 \\
-1 &  2 & -1&  0 &  0 &  0 &  0 & 0 \\
 0 & -1 &  2 & -1 &  0 &  0 &  0 & 0 \\
 0 &  0 & -1 &  2 & -1 &  0 &  0 & 0 \\
 0 &  0 &  0 & -1 &  2 & -1 &  0 & -1 \\
 0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 \\
 0 &  0 &  0 &  0 &  0 & -1 &  2 & 0 \\
 0 &  0 & 0 &  0 &  -1 &  0 &  0 & 2
\end{array}\right ].</math>
The [[determinant]] of this matrix is equal to 1.

=== Simple roots ===
[[File:E8HassePoset.svg|thumb|upright|[[Hasse diagram]] of E<sub>8</sub> [[Root system#The root poset|root poset]] with edge labels identifying added simple root position]]

A set of [[Simple root (root system)|simple root]]s for a root system Φ is a set of roots that form a [[basis (linear algebra)|basis]] for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.

Given the E<sub>8</sub> [[Cartan matrix]] (above) and a [[Dynkin diagram]] node ordering of: [[File:DynkinE8.svg|150px]]

One choice of [[Simple root (root system)|simple root]]s is given by the rows of the following matrix:
: <math>\left [\begin{array}{rr}
1&-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&1&-1&0&0&0 \\
0&0&0&0&1&-1&0&0 \\
0&0&0&0&0&1&1&0 \\
-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
0&0&0&0&0&1&-1&0 \\
\end{array}\right ].</math>

With this numbering of nodes in the Dynkin diagram, the highest root in the root system has [[Coxeter label]]s (2, 3, 4, 5, 6, 4, 2, 3). Using this representation of the simple roots, the lowest root is given by
: <math>\left [\begin{array}{rr}
-1&0&0&0&0&0&0&1 \\
\end{array}\right ].</math>
The only simple root that can be added to the lowest root to obtain another root is the one corresponding to node 1 in this labeling of the Dynkin diagram — as is to be expected from the [[Dynkin diagram#Affine Dynkin diagrams|affine Dynkin diagram]] for <math>{\tilde{\mathrm{E}}}_{8}</math>. The [[Hasse diagram]] to the right enumerates the 120 roots of positive height relative to any particular choice of simple roots consistent with this node numbering.

Note that the Hasse diagram does not represent the full Lie algebra, or even the full root system. The 120 roots of negative height relative to the same set of simple roots can be adequately represented by a second copy of the Hasse diagram with the arrows reversed; but it is less straightforward to connect these two diagrams via a basis for the eight-dimensional Cartan subalgebra. In the notation of [[Semisimple Lie algebra#Structure|the exposition of Chevalley generators and Serre relations]]: Insofar as an arrow represents the Lie bracket by the generator <math>e_i</math> associated with a simple root, each root in the height -1 layer of the reversed Hasse diagram must correspond to some <math>f_i</math> and can have only one upward arrow, connected to a node in the height 0 layer representing the element of the Cartan subalgebra given by <math>h_i = [e_i, f_i]</math>. But the upward arrows from the height 0 layer must then represent <math>[e_i, h_j] = -a_{ji} e_i</math>, where <math>a_{ji}</math> is (the transpose of) the Cartan matrix. One could draw multiple upward arrows from each <math>h_j</math> associated with all <math>e_i</math> for which <math>[e_i, h_j]</math> is nonzero; but this neither captures the numerical entries in the Cartan matrix nor reflects the fact that each <math>e_i</math> only has nonzero Lie bracket with one degree of freedom in the Cartan subalgebra (just not the same degree of freedom as <math>h_i</math>).

More fundamentally, this organization implies that the ''span'' of the generators designated as "the" Cartan subalgebra is somehow inherently special, when in most applications, any mutually commuting set of eight of the 248 Lie algebra generators (of which there are many!) — or any eight linearly independent, mutually commuting Lie derivations on any manifold with E<sub>8</sub> structure — would have served just as well. Once a Cartan subalgebra has been selected (or defined ''a priori'', as in the case of a lattice), a basis of "Cartan generators" (the <math>h_i</math> among the Chevalley generators) and a root system are a useful way to describe structure ''relative to this subalgebra''. But the root system map is not the Lie algebra (let alone group!) territory. Given a set of Chevalley generators, most degrees of freedom in a Lie algebra and their sparse Lie brackets with <math>{e_i}</math> can be represented schematically as circles and arrows, but this simply breaks down on the chosen Cartan subalgebra. Such are the hazards of schematic visual representations of mathematical structures.

=== Weyl group ===
The [[Weyl group]] of E<sub>8</sub> is of order 696729600, and can be described as O{{su|p=+|b=8}}(2): it is of the form 2.''G''.2 (that is, a [[Schur multiplier|stem extension]] by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group ''G'') where ''G'' is the unique [[simple group]] of order 174182400 (which can be described as  PSΩ<sub>8</sub><sup>+</sup>(2)).<ref>{{citation |last1=Conway |first1=John Horton |author-link1=John Horton Conway |last2=Curtis |first2=Robert Turner |last3=Norton |first3=Simon Phillips |author-link3=Simon P. Norton |last4=Parker |first4=Richard A |author-link4=Richard A. Parker |last5=Wilson |first5=Robert Arnott |author-link5=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|Atlas of Finite Groups]]: Maximal Subgroups and Ordinary Characters for Simple Groups |year=1985 |publisher=Oxford University Press |isbn=0-19-853199-0 |page=85 }}</ref>

=== E<sub>8</sub> root lattice ===
{{main|E8 lattice|l1=E<sub>8</sub> lattice}}

The integral span of the E<sub>8</sub> root system forms a [[lattice (group)|lattice]] in '''R'''<sup>8</sup> naturally called the '''[[E8 lattice|E<sub>8</sub> root lattice]]'''. This lattice is rather remarkable in that it is the only (nontrivial) even, [[unimodular lattice]] with rank less than 16.

=== Simple subalgebras of E<sub>8</sub> ===
[[File:E8SubgroupTree.svg|thumb|An incomplete simple subgroup tree of E<sub>8</sub>]]

The Lie algebra E<sub>8</sub> contains as subalgebras all the [[exceptional Lie algebra]]s as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.

== Chevalley groups of type E<sub>8</sub> ==
{{harvtxt|Chevalley|1955}} showed that the points of the (split) algebraic group E<sub>8</sub> (see [[#E8 as an algebraic group|above]]) over a [[finite field]] with ''q'' elements form a finite [[Group of Lie type|Chevalley group]], generally written E<sub>8</sub>(''q''), which is simple for any ''q'',<ref>{{citation |first=Roger W. |last=Carter |title=Simple Groups of Lie Type |author-link=Roger Carter (mathematician) |publisher=John Wiley & Sons |series=Wiley Classics Library |isbn=0-471-50683-4 |year=1989}}</ref><ref>{{citation |first=Robert A. |last=Wilson |title=The Finite Simple Groups |author-link=Robert Arnott Wilson |publisher=[[Springer-Verlag]] |series=[[Graduate Texts in Mathematics]] |volume=251 |isbn=978-1-84800-987-5 |year=2009}}</ref> and constitutes one of the infinite families addressed by the [[classification of finite simple groups]].  Its number of elements is given by the formula {{OEIS|id=A008868}}:
: <math>q^{120}\left(q^{30} - 1\right)\left(q^{24} - 1\right)\left(q^{20} - 1\right)\left(q^{18} - 1\right)\left(q^{14} - 1\right)\left(q^{12} - 1\right)\left(q^8 - 1\right)\left(q^2 - 1\right)</math>

The first term in this sequence, the order of E<sub>8</sub>(2), namely {{gaps|337|804|753|143|634|806|261|388|190|614|085|595|079|991|692|242|467|651|576|160|959|909|068|800|000}} ≈ {{val|3.38|e=74}}, is already larger than the size of the [[Monster group]]. This group E<sub>8</sub>(2) is the last one described (but without its character table) in the [[ATLAS of Finite Groups]].<ref>Conway &amp;al, ''op. cit.'', p.&nbsp;235.</ref>

The [[Schur multiplier]] of E<sub>8</sub>(''q'') is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order ''f'' if {{nowrap|1=''q'' = ''p<sup>f</sup>''}}, where ''p'' is prime).

{{harvtxt|Lusztig|1979}} described the unipotent representations of finite groups of type ''E''<sub>8</sub>.

== Subgroups, subalgebras, and extensions ==
The smaller exceptional groups [[E7 (mathematics)|E<sub>7</sub>]] and [[E6 (mathematics)|E<sub>6</sub>]] sit inside E<sub>8</sub>. In the compact group, both E<sub>6</sub>×SU(3)/('''Z'''{{hsp}}/{{hsp}}3'''Z''') and E<sub>7</sub>×SU(2)/(+1,−1) are [[maximal subgroup]]s of E<sub>8</sub>.

[[File:E8 Maximal Embeddings.svg|thumb|300px|Embeddings of the maximal subgroups of E<sub>8</sub> up to dimension 248 with associated projection matrix.]]

The 248-dimensional adjoint representation of E<sub>8</sub> may be considered in terms of its [[restricted representation]] to the first of these subgroups. It transforms under E<sub>6</sub>×SU(3) as a sum of [[tensor product representation]]s, which may be labelled as a pair of dimensions as (78,1) + (1,8) + ({{overline|27}},3) + (27,{{overline|3}}). (Since the maximal subgroup is actually the quotient of this group product by a finite group, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.) Since the adjoint representation can be described by the roots together with the generators in the [[Cartan subalgebra]], we may choose a particular E<sub>6</sub> root system within E<sub>8</sub> and decompose the sum representation relative to this E<sub>6</sub>. In this description,
* (78,1), a copy of <math>\mathfrak{e}_6</math>, consists of the 72 roots with (−{{frac|1|2}},−{{frac|1|2}},−{{frac|1|2}}), (0,0,0), or ({{frac|1|2}},{{frac|1|2}},{{frac|1|2}}) in the last three dimensions, together with five Cartan generators corresponding to the first five dimensions and one Cartan generator corresponding to an equally weighted sum of the last three Cartan generators of the E<sub>8</sub> system;
* (1,8), a copy of <math>\mathfrak{su}(3)</math>, consists of the six roots with permutations of (0,1,−1) in the last three dimensions, together with two Cartan generators corresponding to the two trace-free combinations of the last three Cartan generators of the E<sub>8</sub> system;
* (<u style="text-decoration:overline">27</u>,3) consists of all roots with permutations of (-1,0,0), (−{{frac|1|2}},{{frac|1|2}},{{frac|1|2}}), or (0,1,1) in the last three dimensions; and
* (27,<u style="text-decoration:overline">3</u>) consists of all roots with permutations of (1,0,0), ({{frac|1|2}},−{{frac|1|2}},−{{frac|1|2}}), or (0,-1,-1) in the last three dimensions.

The 248-dimensional adjoint representation of E<sub>8</sub>, when similarly restricted to the second maximal subgroup, transforms under E<sub>7</sub>×SU(2) as: (133,1) + (1,3) + (56,2). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description,
* (133,1), a copy of <math>\mathfrak{e}_7</math>, consists of the 126 roots with (−1,−1), (−{{frac|1|2}},−{{frac|1|2}}), (0,0), ({{frac|1|2}},{{frac|1|2}}), or (1,1) in the last two dimensions, together with six Cartan generators corresponding to the first six dimensions and one Cartan generator corresponding to an equally weighted sum of the last two Cartan generators of the E<sub>8</sub> system;
* (1,3), a copy of <math>\mathfrak{su}(2)</math>, consists of the two roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1), together with one Cartan generator corresponding to the trace-free combination of the last two Cartan generators of the E<sub>8</sub> system; and
* (56,2) consists of all roots with permutations of (0,-1), ({{frac|1|2}},−{{frac|1|2}}), or (1,0) in the last two dimensions.

The connection between these two descriptions is given by the graded exceptional Lie algebra constructions of [[Kantor–Koecher–Tits construction|J. Tits]] and [[Structurable algebra|B. N. Allison]].  Any 27-dimensional representation of E<sub>6</sub> can be equipped with a non-associative (but strictly [[power-associative]]) Jordan product operation to form an [[Albert algebra]] (an important exceptional case in [https://www.math.uni-bielefeld.de/LAG/man/089.pdf algebraic constructions]). The [[Kantor–Koecher–Tits construction]] applied to this Albert algebra recovers the 78-dimensional <math>\mathfrak{e}_6</math> as the reduced structure algebra of the Albert algebra. This <math>\mathfrak{e}_6</math>, together with the 27 and <u style="text-decoration:overline">27</u> representations and the grade operator (the element of the Cartan subalgebra with weight -1 on the 27, +1 on the <u style="text-decoration:overline">27</u>, and 0 on the 78), forms an <math>\mathfrak{e}_7</math> 3-graded Lie algebra.  A complete exposition of this construction may be found in standard texts on Jordan algebras such as [https://www.worldcat.org/title/structure-and-representations-of-jordan-algebras/oclc/1077 Jacobson 1968] or [http://old.math.nsc.ru/LBRT/a1/files/mccrimmon.pdf McCrimmon 2004].

Starting this 3-graded Lie algebra construction with any particular 27-dimensional representation, embedded within <math>\mathfrak{e}_8</math>, of any particular E<sub>6</sub> subgroup of E<sub>8</sub> produces the corresponding <math>\mathfrak{e}_7</math> subalgebra.  The particular <math>\mathfrak{e}_7</math> in the E<sub>7</sub>×SU(2) decomposition given above corresponds to choosing the 27 consisting of all roots with (1,0,0), ({{frac|1|2}},-{{frac|1|2}},-{{frac|1|2}}), or (0,-1,-1) in the last three dimensions (in order), with the grade operator having weight (-1,{{frac|1|2}},{{frac|1|2}}) in these dimensions; or equivalently to choosing the "<u style="text-decoration:overline">27</u>" consisting of all roots with (-1,0,0), (−{{frac|1|2}},{{frac|1|2}},{{frac|1|2}}), or (0,1,1) in the last three dimensions (in order), with the grade operator having weight (1,-{{frac|1|2}},-{{frac|1|2}}) in these dimensions.  Note that there is nothing special about this choice of dimensions — the <math>\mathbb R^{8}</math> within which the root system is embedded is not the set of eight independent but non-orthogonal axes corresponding to the simple roots, and any three dimensions will do — and there are also constructions using other equivalent groupings of roots.  What matters is that the kernel of the Lie bracket with the generator chosen as the "grade operator" be an <math>\mathfrak{e}_6</math> subalgebra (plus a central <math>\mathbb{R}^{2}</math> associated with the grade operator itself and the remaining generator of the Cartan subalgebra), not <math>\mathfrak{a}_6 \oplus \mathbb{R}^{2}</math>, <math>\mathfrak{a}_7 \oplus \mathbb{R}</math>, <math>\mathfrak{e}_7 \oplus \mathbb{R}</math>, ''etc.''

(In the simple Lie algebra case, the sign of the grade of the 27 versus the <u style="text-decoration:overline">27</u> representation is a matter of convention, as is the scale of the grading. However, choosing -1 as the grade of the 27-dimensional "vector" representation is consistent with an extension of the 3-graded algebra to higher positive grades via the exterior algebra over the <u style="text-decoration:overline">27</u> "covector" representation. The "vector" representation then lies, not in this nonnegative-graded exterior algebra, but in the graded algebra of derivations over the exterior algebra; the 78-dimensional <math>\mathfrak{e}_6</math> is the grade 0 subalgebra (a subalgebra of the inner derivations by "vector-valued 1-forms") of this graded algebra of derivations. For details on how this asymmetric structure works starting from a general 3-graded algebra, see references at [[Frölicher–Nijenhuis bracket]]. The relevance of this observation to E<sub>8</sub> is simply that E<sub>7</sub> and E<sub>8</sub> are their own clusters of structures, distinguished as exceptional simple Lie groups/algebras, and that any particular reconstruction of them using representations of their subgroups/subalgebras will have extensions beyond the motivating case.  Varying conventions of sign, scale, and conjugate relationship in the literature are due not just to inaccuracies but also to the directions in which the authors seek to extend their constructions.)

The distinguished <math>\mathfrak{su}(2)</math> in the E<sub>7</sub>×SU(2) decomposition above is then given by the subalgebra of <math>\mathfrak{su}(3)</math> that commutes with the grade operator (which lies in the Cartan subalgebra of this <math>\mathfrak{su}(3)</math>).  Of the four remaining roots in the <math>\mathfrak{su}(3)</math>, two are of grade {{frac|1|2}} and two are of grade -{{frac|1|2}}.  In the convention where the 27 of E<sub>6</sub> used to construct the <math>\mathfrak{e}_7</math> has grade -1 and the <u style="text-decoration:overline">27</u> has grade +1, the other two 27's have grade +{{frac|1|2}} and the other two <u style="text-decoration:overline">27</u>'s have grade -{{frac|1|2}}, as is apparent from permuting the values of the last three roots in the description above.  Grouping these 4×(1+27)=112 generators to form the grade +{{frac|1|2}} and -{{frac|1|2}} subspaces of <math>\mathfrak{e}_8</math> (relative to the original choice of grade operator within <math>\mathfrak{e}_7</math>), each subspace may be given a quite particular non-associative (nor even power-associative) product operation, resulting in two copies of [https://www.jstor.org/stable/71948 Brown's 56-dimensional structurable algebra].  Allison's 5-graded Lie algebra construction based on this [[structurable algebra]] recovers the original <math>\mathfrak{e}_8</math>.  (Allison's 5-grading differs from the above by a factor -2.) Grouping these generators differently, based on their weights relative to the Cartan generator of the <math>\mathfrak{su}(2)</math> ''orthogonal'' to <math>\mathfrak{e}_7</math>, gives two 56-dimensional subspaces that each carry the lowest-dimensional non-trivial irreducible representation of E<sub>7</sub>.  Either of these may be combined with the Cartan generator to form a 57-dimensional [[Heisenberg algebra]], and adjoining this to <math>\mathfrak{e}_7</math> produces the (non-simple) Lie algebra [[E7 1/2|E<sub>7 1/2</sub>]] described by Landsberg and Manivel.

From the perspective in which the 27-dimensional grade -1 subspace of <math>\mathfrak{e}_8</math> (relative to a choice of grade operator) plays the role of "vector" representation of E<sub>6</sub> and the <u style="text-decoration:overline">27</u> with roots opposite it plays the role of "covector" representation, it is natural to look for "spinor" representations in the grade +{{frac|1|2}} and -{{frac|1|2}} subspaces, or in some other combination of the (<u style="text-decoration:overline">27</u>,3) and (27,<u style="text-decoration:overline">3</u>) representations of E<sub>6</sub>×SU(3), and to attempt to relate these to geometrical spinors in the [[Clifford algebra]] sense as employed in [[quantum field theory]]. Variations on this idea are [https://arxiv.org/pdf/2005.03048.pdf common in the physics literature]. See [https://arxiv.org/pdf/0905.2658.pdf Distler and Garibaldi 2009] for discussion of the mathematical obstacles to constructing a ''chiral'' gauge theory based on E<sub>8</sub>. The structure of <math>\mathfrak{e}_8</math> relative to its <math>\mathfrak{e}_6</math> subalgebra, together with the conventional scaling of elements of the Cartan subalgebra, invites extensions by geometric analogy but does not necessarily imply a relationship to low-dimensional geometry or low-energy physics. The same may be said of connections to Jordan and Heisenberg algebras, whose historical origins are intertwined with the development of quantum mechanics. Not every [[The Treachery of Images|visual representation evocative of a tobacco pipe]] will hold tobacco.

The finite quasisimple groups that can embed in (the compact form of) E<sub>8</sub> were found by {{harvtxt|Griess|Ryba|1999}}.

The [[Dempwolff group]] is a subgroup of (the compact form of) E<sub>8</sub>. It is contained in the [[Thompson sporadic group]], which acts on the underlying vector space of the Lie group E<sub>8</sub> but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E<sub>8</sub>('''F'''<sub>3</sub>).

The embeddings of the maximal subgroups of E<sub>8</sub> up to dimension 248 are shown to the right.

== Applications ==
The E<sub>8</sub> Lie group has applications in [[theoretical physics]] and especially in [[string theory]] and [[supergravity]]. E<sub>8</sub>×E<sub>8</sub> is the [[gauge group]] of one of the two types of [[heterotic string]] and is one of two [[anomaly (physics)|anomaly-free]] gauge groups that can be coupled to the ''N''&nbsp;= 1 supergravity in ten dimensions. E<sub>8</sub> is the [[U-duality]] group of supergravity on an eight-torus (in its split form).

One way to incorporate the [[standard model]] of particle physics into heterotic string theory is the [[Spontaneous symmetry breaking|symmetry breaking]] of E<sub>8</sub> to its maximal subalgebra SU(3)&times;E<sub>6</sub>.

In 1982, [[Michael Freedman]] used the [[E8 lattice|E<sub>8</sub> lattice]] to construct an example of a [[Topological manifold|topological]] [[4-manifold]], the [[E8 manifold|E<sub>8</sub> manifold]], which has no [[Differential structure|smooth structure]].

[[Antony Garrett Lisi]]'s incomplete "[[An Exceptionally Simple Theory of Everything]]" attempts to describe all known [[fundamental interaction]]s in physics as part of the E<sub>8</sub> Lie algebra.<ref name="SciAm">{{cite journal |doi=10.1038/scientificamerican1210-54 |author1=A. G. Lisi |author-link1=Antony Garrett Lisi |author2=J. O. Weatherall |author-link2=<!--James Owen Weatherall--> |year=2010 |title=A Geometric Theory of Everything |journal=[[Scientific American]] |volume=303 |issue=6 |pages=54–61 |url=http://www.scientificamerican.com/article.cfm?id=a-geometric-theory-of-everything |pmid=21141358|bibcode=2010SciAm.303f..54L }}</ref><ref name="seed">{{cite news |author= Greg Boustead |title=Garrett Lisi's Exceptional Approach to Everything |url=http://seedmagazine.com/news/2008/11/garrett_lisis_exceptional_appr.php |archive-url=https://web.archive.org/web/20090202122533/http://www.seedmagazine.com/news/2008/11/garrett_lisis_exceptional_appr.php |url-status=unfit |archive-date=2009-02-02 |work=SEED Magazine |date=2008-11-17}}</ref>

{{Harvs|txt |last1=Coldea |first1=R. |last2=Tennant |first2=D. A. |last3=Wheeler |first3=E. M. |last4=Wawrzynska |first4=E. |last5=Prabhakaran |first5=D. |last6=Telling |first6=M. |last7=Habicht |first7=K. |last8=Smeibidl |first8=P. |last9=Kiefer |first9=K. |title=Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E<sub>8</sub> Symmetry |doi=10.1126/science.1180085 |year=2010 |journal=[[Science (journal)|Science]] |volume=327 |issue=5962 |pages=177–180}} reported an experiment where the [[electron spin]]s of a [[cobalt]]-[[niobium]] crystal exhibited, under certain conditions, two of the eight peaks related to E<sub>8</sub> that were predicted by {{harvtxt|Zamolodchikov|1989}}.<ref>{{Cite web|first = David |last = Shiga|title='Most beautiful' math structure appears in lab for first time|url=https://www.newscientist.com/article/dn18356-most-beautiful-math-structure-appears-in-lab-for-first-time/ |date = 2010-01-07 |access-date=2023-02-01|website=New Scientist|language=en-US}}</ref><ref>[https://www.ams.org/notices/201108/rtx110801055p.pdf Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?], ''[[Notices of the American Mathematical Society]]'', September 2011.</ref>

== History ==
{{harvs|txt |authorlink=Wilhelm Killing |first=Wilhelm |last=Killing |year1=1888a |year2=1888b |year3=1889 |year4=1890}} discovered the complex Lie algebra  E<sub>8</sub>  during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by [[Élie Cartan]]. Cartan determined that a complex simple Lie algebra of type E<sub>8</sub> admits three real forms. Each of them gives rise to a simple [[Lie group]] of dimension 248, exactly one of which (as for any complex simple Lie algebra) is [[compact Lie group|compact]]. {{harvtxt|Chevalley|1955}} introduced [[algebraic group]]s and Lie algebras of type E<sub>8</sub> over other [[field (mathematics)|fields]]: for example, in the case of [[finite field]]s they lead to an infinite family of [[finite simple group]]s of Lie type. E<sub>8</sub> continues to be an area of active basic research by [[Atlas of Lie Groups and Representations]], which aims to determine the unitary representations of all the Lie groups.<ref>{{Cite web|url=https://aimath.org/E8/|title=AIM math: Representations of E8|website=aimath.org}}</ref>

== See also ==
* [[En (Lie algebra)|E<sub>''n''</sub>]]

== Footnotes ==
{{Reflist}}

== References ==
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* {{citation |last=Baez |first=John C. |author-link=John C. Baez |title=The octonions |url=https://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html |doi=10.1090/S0273-0979-01-00934-X  |mr=1886087 |year=2002 |journal=[[Bulletin of the American Mathematical Society]] |series=New Series |volume=39 |issue=2 |pages=145–205|arxiv=math/0105155 |s2cid=586512 }}
* {{citation |last=Chevalley |first=Claude |author-link=Claude Chevalley |title=Sur certains groupes simples |url=http://projecteuclid.org/euclid.tmj/1178245104 |doi=10.2748/tmj/1178245104 |mr=0073602 |year=1955 |journal=The Tohoku Mathematical Journal |series=Second Series |issn=0040-8735 |volume=7 |issue=1–2 |pages=14–66|doi-access=free }}
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* {{citation |last1=Griess |first1=Robert L. |last2=Ryba |first2=A. J. E. |title=Finite simple groups which projectively embed in an exceptional Lie group are classified! |url=https://www.ams.org/bull/1999-36-01/S0273-0979-99-00771-5/home.html |mr=1653177 |year=1999 |journal=Bulletin of the American Mathematical Society |series=New Series |volume=36 |issue=1 |pages=75–93 |doi=10.1090/S0273-0979-99-00771-5|doi-access=free }}
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== External links ==
=== Lusztig–Vogan polynomial calculation ===
* [http://www.liegroups.org/ Atlas of Lie groups]
* [http://www.liegroups.org/kle8.html Kazhdan–Lusztig–Vogan Polynomials for E<sub>8</sub>]
* [http://atlas.math.umd.edu/kle8.narrative.html Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E<sub>8</sub>]
* {{citation |author=[[American Institute of Mathematics]] |title=Mathematicians Map E<sub>8</sub> |date=March 2007 |url=http://aimath.org/E8/}}
* [http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html The ''n''-Category Café], a [[University of Texas]] blog posting by [[John Baez]] on E<sub>8</sub>.

=== Other links ===
* [http://www-math.mit.edu/~dav/e8plane.html Graphic representation of E<sub>8</sub> root system].
* The list of dimensions of [[irreducible representation]]s of the complex form of E<sub>8</sub> is sequence [[OEIS:A121732|A121732]] in the [[OEIS]].

{{Exceptional Lie groups}}
{{String theory topics |state=collapsed}}

[[Category:E8 (mathematics)| ]]
[[Category:Exceptional Lie algebras]]
[[Category:Lie groups]]