# E₈

In

mathematics ,**E**is the name given to a family of closely related structures. In particular, it is the name of four exceptional simple_{8}Lie algebra s as well as that of the six associated simpleLie group s. It is also the name given to the correspondingroot system ,root lattice , and Weyl/Coxeter group , and to some finite simpleChevalley group s. E_{8}was discovered between the years of 1888 and 1890 byWilhelm Killing , though he did not prove its existence, which was first shown by E. Cartan.The designation E

_{8}comes from Wilhelm Killing andÉlie Cartan 's classification of the complexsimple Lie algebra s, which fall into four infinite families 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_{8}algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.**Basic description**E

_{8}has rank 8 and dimension 248 (as amanifold ). The vectors of the root system are in eight dimensions and are specified later in this article. TheWeyl group of E_{8}, which acts as asymmetry group of the maximal torus by means of theconjugation operation from the whole group, is of order 696729600.E

_{8}is unique among simple Lie groups in that its non-trivial representation of smallest dimension is theadjoint representation (of dimension 248) acting on the Lie algebra E_{8}itself.There is a Lie algebra E

_{n}for every integer "n"≥3, which is infinite dimensional if "n" is greater than 8.**Real forms**The complex Lie group E

_{8}ofcomplex dimension 248 can be considered as a simple real Lie group of (real) dimension 496, which is simply connected, has maximal compact subgroup the compact form of E_{8}, and has an outer automorphism group of order 2 generated by complex conjugation.As well as the complex Lie group of type E

_{8}, there are three real forms of the group, all of real dimension 248, as follows:

*A compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.

*A split form, which has maximal compact subgroup Spin(16)/(**Z**/2**Z**), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.

*A third form, which has maximal compact subgroup E_{7}×SU(2)/(−1×−1), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.For a complete list of real forms of simple Lie algebras, see the

list of simple Lie groups .**Representation theory**The coefficients of the character formulas for infinite dimensional irreducible representations of E

_{8}depend on some large square matrices consisting of polynomials, theLusztig–Vogan polynomial s, an analogue ofKazhdan–Lusztig polynomial s introduced forreductive group s in general byGeorge Lusztig andDavid Vogan (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 group of 18 mathematicians and computer scientists, led by Jeffrey Adams, with much of the programming done by

Fokko du Cloux . The most difficult case (for exceptional groups) is the splitreal form of E_{8}(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"_{8}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.**Constructions**One can construct the (compact form of the) E

_{8}group as theautomorphism group of the corresponding**e**_{8}Lie algebra. This algebra has a 120-dimensional subalgebra**so**(16) generated by "J"_{"ij"}as well as 128 new generators "Q"_{"a"}that transform as aWeyl-Majorana spinor of**spin**(16). These statements determine the commutators:$[J\_\{ij\},J\_\{kell\}]\; =delta\_\{jk\}J\_\{iell\}-delta\_\{jell\}J\_\{ik\}-delta\_\{ik\}J\_\{jell\}+delta\_\{iell\}J\_\{jk\}$

as well as

:$[J\_\{ij\},Q\_a]\; =\; frac\; 14\; (gamma\_igamma\_j-gamma\_jgamma\_i)\_\{ab\}\; Q\_b,$

while the remaining commutator (not anticommutator!) is defined as

:$[Q\_a,Q\_b]\; =gamma^\{\; [i\}\_\{ac\}gamma^\{j]\; \}\_\{cb\}\; J\_\{ij\}.$

It is then possible to check that the

Jacobi identity is satisfied.**Geometry**The compact real form of E

_{8}is theisometry group of a 128-dimensionalRiemannian manifold known informally as the 'octo-octonionic projective plane' because it can be built using an algebra that is the tensor product of theoctonion s with themselves. This can be seen systematically using a construction known as the "magic square", due toHans Freudenthal andJacques Tits (see J.M. Landsberg, L. Manivel, (2001)).**E**_{8}root systemA

root system of rank "r" is a particular finite configuration of vectors, called "roots", which span an "r"-dimensionalEuclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.The

**E**is a rank 8 root system containing 240 root vectors spanning_{8}root system**R**^{8}. It isirreducible in the sense that it cannot be built from root systems of smaller rank. Each of the root vectors in E_{8}have equal length. It is convenient for many purposes to normalize them to have length √2.**Construction**In the so-called "even coordinate system" E

_{8}is given as the set of all vectors in**R**^{8}with length squared equal to 2 such that coordinates are either allinteger s or allhalf-integer s and the sum of the coordinates is even.Explicitly, there are 112 roots with integer entries obtained from:$(pm\; 1,pm\; 1,0,0,0,0,0,0),$by taking an arbitrary combination of signs and an arbitrary

permutation of coordinates, and 128 roots with half-integer entries obtained from:$left(pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12,pm\; frac12\; ight)\; ,$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.The 112 roots with integer entries form a D

_{8}root system. The E_{8}root system also contains a copy of A_{8}(which has 72 roots) as well as E_{6}and E_{7}(in fact, the latter two are usually "defined" as subsets of E_{8}).In the "odd coordinate system" E

_{8}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.**imple roots**A set of

simple root s for a root system Φ is a set of roots that form a 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.One choice of simple roots for E

_{8}(by no means unique) is given by the rows of the following matrix::$left\; [egin\{smallmatrix\}frac\{1\}\{2\}-frac\{1\}\{2\}-frac\{1\}\{2\}-frac\{1\}\{2\}-frac\{1\}\{2\}-frac\{1\}\{2\}-frac\{1\}\{2\}frac\{1\}\{2\}\backslash -11000000\; \backslash 0-1100000\; \backslash 00-110000\; \backslash 000-11000\; \backslash 0000-1100\; \backslash 00000-110\; \backslash 11000000\; \backslash end\{smallmatrix\}\; ight\; ]\; .$**Dynkin diagram**The

Dynkin diagram for E_{8}is given by: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 which are not joined by a line areorthogonal .**Cartan matrix**The

Cartan matrix of a rank "r" root system is an "r" × "r" matrix whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by:$A\_\{ij\}\; =\; 2frac\{(alpha\_i,alpha\_j)\}\{(alpha\_i,alpha\_i)\}$where (-,-) is the Euclideaninner product and "α"_{"i"}are the simple roots. The entries are independent of the choice of simple roots (up to ordering).The Cartan matrix for E

_{8}is given by:$left\; [egin\{smallmatrix\}\; 2\; -1\; 0\; 0\; 0\; 0\; 0\; 0\; \backslash -1\; 2\; -1\; 0\; 0\; 0\; 0\; 0\; \backslash \; 0\; -1\; 2\; -1\; 0\; 0\; 0\; -1\; \backslash \; 0\; 0\; -1\; 2\; -1\; 0\; 0\; 0\; \backslash \; 0\; 0\; 0\; -1\; 2\; -1\; 0\; 0\; \backslash \; 0\; 0\; 0\; 0\; -1\; 2\; -1\; 0\; \backslash \; 0\; 0\; 0\; 0\; 0\; -1\; 2\; 0\; \backslash \; 0\; 0\; -1\; 0\; 0\; 0\; 0\; 2end\{smallmatrix\}\; ight\; ]\; .$Thedeterminant of this matrix is equal to 1.**E**_{8}root latticeThe integral span of the E

_{8}root system forms a lattice in**R**^{8}naturally called the**E**. This lattice is rather remarkable in that it is the only (nontrivial) even,_{8}root latticeunimodular lattice with rank less than 16.**imple subalgebras of E**_{8}The Lie algebra E8 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. Some algebras are more obvious such as SU(n) is a subalgebra of O(2n) and some are less obvious especially the exceptional algebras G2, F4, E6, E7 and E8. Theorthogonal andunitary subalgebras are particularly important in physics as they are used to representspace-time andbosonic symmetries respectively. Some of the smaller algebras are equivalent e.g. O(3)~SU(2).**ubgroups**The smaller exceptional groups E

_{7}and E_{6}sit inside E_{8}. In the compact group, both (E_{7}×SU(2)) / (**Z**/2**Z**) and (E_{6}×SU(3)) / (**Z**/3**Z**) aremaximal subgroup s ofE_{8}.The 248-dimensional adjoint representation of E

_{8}may be considered in terms of itsrestricted representation to the first of these subgroups. It transforms under SU(2)×E_{7}as a sum oftensor product representation s, which may be labelled as a pair of dimensions as:$(3,1)\; +\; (1,133)\; +\; (2,56).\; ,!$(Since there is a quotient in the product, 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 see that decomposition by looking at these. In this description:* The (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension.

* The (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−1/2,−1/2) or (1/2,1/2) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.

* The (2,56) consists of all roots with permutations of (1,0), (−1,0) or (1/2,−1/2) in the last two dimensions.The 248-dimensional adjoint representation of E

_{8}, when similarly restricted, transforms under SU(3)×E_{6}as::$(8,1)\; +\; (1,78)\; +\; (3,27)\; +\; (overline\{3\},overline\{27\}).$

We may again see the decomposition by looking at the roots together with the generators in the

Cartan subalgebra . In this description:* The (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions.

* The (1,78) consists of all roots with (0,0,0), (−1/2,−1/2,−1/2) or (1/2,1/2,1/2) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.

* The (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−1/2,1/2,1/2) in the last three dimensions.

* The (__3__,__27__) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (1/2,−1/2,−1/2) in the last three dimensions.The finite quasisimple groups that can embed in (the compact form of) E

_{8}were found by harvtxt|Griess|Ryba|1999**Applications**The E

_{8}Lie group has applications intheoretical physics , in particular instring theory andsupergravity . The group E_{8}×E_{8}(theCartesian product of two copies of E_{8}) serves as thegauge group of one of the two types ofheterotic string and is one of two anomaly-free gauge groups that can be coupled to the "N" = 1supergravity in 10 dimensions.E_{8}is theU-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 symmetry breaking of E_{8}to its maximal subalgebra SU(3)×E_{6}.In 1982,

Michael Freedman used the E_{8}lattice to construct an example of a topological4-manifold , the E_{8}manifold, which has no smooth structure.In 2007,

Garrett Lisi used E_{8}in hisAn Exceptionally Simple Theory of Everything .**E8 Investigation Tools**These packages can be used to explore E₈ sets:

* [*http://deferentialgeometry.org/epe/ The Elementary Particle Explorer*]

* [*http://theoryofeverything.org/TOE/JGM/e8Flyer.nbp E8Flyer*]

* [*http://www.measurementalgebra.com/E8.html 2D monochrome E8 animated projections*]**ee also***

Classical group

*Lie Algebra **References***

John Frank Adams (1996), "Lectures on Exceptional Lie Groups" (Chicago Lectures in Mathematics), edited by Zafer Mahmud and Mamora Mimura, University of Chicago Press, ISBN 0-226-00527-5.

*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=http://www.ams.org/bull/1999-36-01/S0273-0979-99-00771-5/home.html | id=MathSciNet | id = 1653177 | year=1999 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=36 | issue=1 | pages=75–93

*Killing, "Die Zusammensetzung der stetigen/endlichen Transformationsgruppen" Mathematische Annalen, Volume 31, Number 2 June, 1888, pages 252–290 DOI|10.1007/BF01211904, Volume 33, Number 1 March, 1888, pages 1–48 DOI|10.1007/BF01444109, Volume 34, Number 1 March, 1889, pages 57–122 DOI|10.1007/BF01446792, Volume 36, Number 2 June, 1890, pages 161–189 DOI|10.1007/BF01207837

*J.M. Landsberg and L. Manivel (2001), "The projective geometry of Freudenthal's magic square", Journal of Algebra, Volume 239, Issue 2, pages 477–512, doi|10.1006/jabr.2000.8697, [*http://www.arxiv.org/abs/math/9908039 arXiv:math/9908039v1*] .

*Citation

last1=Lusztig

first1=George

author1-link=George Lusztig

last2=Vogan

first2=David

author2-link=David Vogan

title=Singularities of closures of K-orbits on flag manifolds.

year=1983

month=

volume=71

number=2

journal=Inventiones Mathematicae

publisher=Springer-Verlag

pages=365–379

issn=0020-9910

doi=10.1007/BF01389103**External links**Links related to the calculation of the Lusztig-Vogan polynomials.

* [*http://www.liegroups.org/ atlas of Lie groups*]

* [*http://www.liegroups.org/kle8.html Kazhdan-Lusztig-Vogan Polynomials for E*]_{8}

*D. Vogan, [*http://atlas.math.umd.edu/kle8.narrative.html Narrative of the Project to compute Kazhdan-Lusztig Polynomials for E*]_{8}

*D. Vogan, [*http://math.mit.edu/~dav/E8TALK.pdf "The Character Table for E*] Slides for a popular talk on E_{8}, or How We Wrote Down a 453,060 × 453,060 Matrix and Found Happiness"_{8}.

*

* [*http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html The n-Category Café*] —University of Texas blog posting byJohn Baez on E_{8}Other external links:

*; also available [*http://math.ucr.edu/home/baez/octonions/node19.html here*]

* [*http://www-math.mit.edu/~dav/e8plane.html Graphic representation of E*]_{8}root system

* [*http://theoryofeverything.org/TOE/JGM/e8Flyer.nbp E8Flyer*] - An interactive [*http://theoryofeverything.org/TOE/JGM/ece2.mht E8 investigation tool*] based on the free Mathematica Notebook Player) (* see [*http://theoryofeverything.org/TOE/JGM/E8Favorites.pdf*] for screen shots *)

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2010.*