# E8 (mathematics)
**E8** is the largest of the five exceptional simple Lie algebras (and the corresponding Lie groups), a 248-dimensional structure of continuous symmetry with rank 8 and a root system of 240 vectors in eight-dimensional space.[1](https://math.ucr.edu/home/baez/kostant/summary.html) It stands outside the four infinite families of classical Lie algebras and sits at the top of the exceptional chain \( G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8 \).[2](https://arxiv.org/abs/1605.01721)
| Key fact | Detail |
|---|---|
| Dimension | 248, decomposed as an 8-dimensional Cartan subalgebra plus 240 one-dimensional root spaces[1](https://math.ucr.edu/home/baez/kostant/summary.html) |
| Rank | 8, the dimension of the maximal torus[1](https://math.ucr.edu/home/baez/kostant/summary.html) |
| Root system | 240 roots of squared length 2 in \( \mathbb{R}^8 \), with all-integer or all-half-integer coordinates summing to an even integer[3](https://admin.aimath.org/E8/e8.html) |
| Place in classification | Largest of the five exceptional simple Lie algebras; no simple Lie algebras of type \( E_n \) exist for \( n > 8 \)[2](https://arxiv.org/abs/1605.01721) • [4](https://math.stackexchange.com/questions/2489920/exceptional-lie-algebras-e8) |
| Discovery | First sighted in Wilhelm Killing's classification work, published from 1888; Élie Cartan later gave the first complete, rigorous classification[5](https://www.treccani.it/enciclopedia/l-ottocento-matematica-le-origini-della-teoria-dei-gruppi_%28Storia-della-Scienza%29/) |
| Landmark computation | The Kazhdan-Lusztig-Vogan polynomials for E8 were computed on January 8, 2007, in a calculation requiring more than 200 gigabytes of RAM[6](https://aimath.org/e8/computerdetails.html) |
| Physics roles | Gauge group \( E_8 \times E_8 \) of ten-dimensional heterotic string theory; the split real form plays a role in M-theory[7](https://www.patternfieldtheory.com/papers/e8_on_the_aol.pdf) • [8](https://www.aimath.org/E8/technicaldetails.html) |
## Structure and root system
A simple [Lie algebra](https://www.edgepedia.org/lie-algebra) decomposes into root spaces, one for each root, together with a [Cartan subalgebra](https://www.edgepedia.org/cartan-subalgebra) whose dimension is the rank. For E8 the Cartan subalgebra is 8-dimensional and there are 240 roots, giving \( 240 + 8 = 248 \) dimensions in total.[1](https://math.ucr.edu/home/baez/kostant/summary.html) The roots are vectors \( (a_1, a_2, \ldots, a_8) \) in which all coordinates are integers or all are integers plus \( 1/2 \), the coordinate sum is an even integer, and the sum of squares is 2.[3](https://admin.aimath.org/E8/e8.html) Concretely, the 240 roots split as 112 vectors of the form \( \pm(e_i \pm e_j) \) and 128 half-integer vectors, so the complex dimension is \( 8 + 240 = 248 \).[9](https://citeseerx.ist.psu.edu/document?doi=fc62893491fa6dd59d685e091dc776687dc1a445&repid=rep1&type=pdf)
A standard construction, presented in James Humphreys' *Introduction to Lie Algebras and Representation Theory*, builds the root system from an orthonormal basis \( e_1, \ldots, e_8 \) of \( \mathbb{R}^8 \): one takes the lattice generated by the basis vectors and the half-sum \( \delta = \tfrac{1}{2}(e_1 + \cdots + e_8) \), restricts to elements whose integer coefficients sum to an even number, and defines \( \Phi \) as the vectors of squared length 2 in that lattice.[10](https://math.stackexchange.com/questions/3630935/construction-of-the-e-8-root-system)
The algebra is also notably noncommutative relative to its rank: the ratio of dimension to squared rank, \( 248/8^2 \), is the largest among the simple Lie groups.[12](https://math.mit.edu/~gyuri/papers/e8final.pdf)
## History
E8 first appeared in the classification of the complex simple Lie algebras begun by [Wilhelm Killing](https://www.edgepedia.org/wilhelm-killing), a German mathematician who sent Klein a four-part paper starting in 1888 classifying the finite-dimensional complex simple Lie algebras. Killing's work was incomplete, with laborious calculations omitted and some cases not considered, but he published it despite its imperfections.[5](https://www.treccani.it/enciclopedia/l-ottocento-matematica-le-origini-della-teoria-dei-gruppi_%28Storia-della-Scienza%29/) The French mathematician Arthur Tresse brought Killing's work to the attention of his friend [Élie Cartan](https://www.edgepedia.org/elie-cartan), who carried it through to completion: Cartan's thesis contained the first exhaustive classification of the complex semisimple Lie algebras, justifying Killing's intuitions, clarifying his arguments, and correcting his errors and omissions.[5](https://www.treccani.it/enciclopedia/l-ottocento-matematica-le-origini-della-teoria-dei-gruppi_%28Storia-della-Scienza%29/)
## Constructions
Several independent constructions of E8 are known. [Jacques Tits](https://www.edgepedia.org/jacques-tits) introduced in the 1960s a unified construction of the exceptional algebras (except \( G_2 \)) from two ingredients: the [Albert algebra](https://www.edgepedia.org/albert-algebra), an exceptional simple [Jordan algebra](https://www.edgepedia.org/jordan-algebra), and a composition algebra.[13](https://www.researchgate.net/publication/394355183_Linear_models_of_the_exceptional_Lie_algebra_mathfrak_e_8) José Figueroa-O'Farrill, a mathematician at the University of Edinburgh, gave a geometric construction of the exceptional Lie algebras \( F_4 \) and \( E_8 \) starting from the round 8- and 15-spheres respectively, inspired by the Killing superalgebra of a supersymmetric supergravity background.[14](https://arxiv.org/abs/0706.2829) In the Clifford-algebra approach of Pierre-Philippe Dechant, the 120 elements of the three-dimensional icosahedral group \( H_3 \) are doubly covered by 240 eight-component objects which, with a reduced inner product, form exactly the E8 root system.[15](https://arxiv.org/abs/1603.04805) In algebraic group theory, the Allison-Faulkner construction gives Lie algebras of type \( E_8 \), and cohomological invariants in degrees 6 and 8 of this construction detect the isotropy rank of the associated groups.[16](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/abs/allisonfaulkner-construction-of-e8/9326CC94236F22D285A091C8941D01C7)
## The Atlas computation
The Atlas of Lie Groups and Representation project aims to compute the irreducible unitary representations (the unitary dual) of any given real [Lie group](https://www.edgepedia.org/lie-group). Its members wrote an algorithm converting the representation theory of real groups, built on deep work of [Harish-Chandra](https://www.edgepedia.org/harish-chandra), Langlands, Kazhdan, Lusztig, Vogan, and others, into computable combinatorics; making these results explicit was a major achievement of Fokko du Cloux.[8](https://www.aimath.org/E8/technicaldetails.html) In spring 2007 the project succeeded in computing the Kazhdan-Lusztig-Vogan polynomials for E8, a calculation completed on January 8, 2007, that needed a computer with more than 200 gigabytes of RAM because of E8's enormous size and complexity.[17](https://aimath.org/~skrantz/Blurbs/atlas.html) • [6](https://aimath.org/e8/computerdetails.html) For any group of rank 7 or less the same computation runs very quickly; E8 marked the step change.[8](https://www.aimath.org/E8/technicaldetails.html) The calculation provides a precise list of Arthur's unipotent representations for E8, perhaps 200 of the 453,060 total.[8](https://www.aimath.org/E8/technicaldetails.html)
## E8 in physics
E8 appears throughout theoretical physics. The gauge group of ten-dimensional [heterotic string theory](https://www.edgepedia.org/heterotic-string-theory) is \( E_8 \times E_8 \), with dimension 248 for each factor, and the split real form of E8 plays a role in M-theory.[7](https://www.patternfieldtheory.com/papers/e8_on_the_aol.pdf) • [8](https://www.aimath.org/E8/technicaldetails.html)
E8 symmetry has also surfaced experimentally. In 2010, *Science* published "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry" (volume 327, pages 177-180), reporting evidence that a one-dimensional quantum magnet near criticality realizes the E8-symmetric integrable field theory studied by [Alexander Zamolodchikov](https://www.edgepedia.org/alexander-zamolodchikov); the model's Hamiltonian shares eigenvectors with the transfer matrices of Onsager's solved two-dimensional [Ising model](https://www.edgepedia.org/ising-model).[19](https://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally)
## Related mathematics
The 248-dimensional Lie algebra can also be given a structure on \( \mathbb{Z}^{248} \) starting from the root system and the Chevalley relations.[20](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/garibaldie8.pdf) Kostant's analysis shows that the 248-dimensional algebra decomposes as a direct sum of 31 copies of the 8-dimensional Cartan algebra, and that the centralizer of a suitable element in E8 is a product of two copies of the [Standard Model](https://www.edgepedia.org/standard-model) gauge group.[1](https://math.ucr.edu/home/baez/kostant/summary.html)
## References
1. [Kostant on E8, John Baez, University of California, Riverside](https://math.ucr.edu/home/baez/kostant/summary.html)
2. [E8, the most exceptional group (arXiv:1605.01721)](https://arxiv.org/abs/1605.01721)
3. [What is E8? American Institute of Mathematics](https://admin.aimath.org/E8/e8.html)
4. [Exceptional Lie algebras E8, Mathematics Stack Exchange](https://math.stackexchange.com/questions/2489920/exceptional-lie-algebras-e8)
5. [L'ottocento: matematica. Le origini della teoria dei gruppi, Treccani, Storia della Scienza](https://www.treccani.it/enciclopedia/l-ottocento-matematica-le-origini-della-teoria-dei-gruppi_%28Storia-della-Scienza%29/)
6. [The E8 calculation, American Institute of Mathematics](https://aimath.org/e8/computerdetails.html)
7. [Standard Lie-theoretic data for the E8 root system](https://www.patternfieldtheory.com/papers/e8_on_the_aol.pdf)
8. [E8 Technical Details, American Institute of Mathematics](https://www.aimath.org/E8/technicaldetails.html)
9. [Representation Theory lecture notes (CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=fc62893491fa6dd59d685e091dc776687dc1a445&repid=rep1&type=pdf)
10. [Construction of the E8 root system, Mathematics Stack Exchange](https://math.stackexchange.com/questions/3630935/construction-of-the-e-8-root-system)
11. [What's the E8 Picture? University of Michigan](https://websites.umich.edu/~jrs/coxplain.html)
12. [E8 notes, MIT (G. Lusztig)](https://math.mit.edu/~gyuri/papers/e8final.pdf)
13. [Linear models of the exceptional Lie algebra e8 (Cabrera, Draper, Garvín et al.)](https://www.researchgate.net/publication/394355183_Linear_models_of_the_exceptional_Lie_algebra_mathfrak_e_8)
14. [A geometric construction of the exceptional Lie algebras F4 and E8, José Figueroa-O'Farrill (arXiv:0706.2829)](https://arxiv.org/abs/0706.2829)
15. [The E8 geometry from a Clifford perspective (arXiv:1603.04805)](https://arxiv.org/abs/1603.04805)
16. [The Allison-Faulkner construction of E8, Canadian Mathematical Bulletin 65 (2022), 686-701](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/abs/allisonfaulkner-construction-of-e8/9326CC94236F22D285A091C8941D01C7)
17. [Atlas of Lie Groups project blurb, American Institute of Mathematics](https://aimath.org/~skrantz/Blurbs/atlas.html)
18. [An Exceptionally Simple Theory of Everything, A. G. Lisi (ResearchGate record)](https://www.researchgate.net/publication/2217412_An_Exceptionally_Simple_Theory_of_Everything)
19. [Has the Lie group E8 really been detected experimentally? MathOverflow, citing Science 327 (5962): 177-180](https://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally)
20. [E8, the most exceptional group (Edinburgh, Garibaldi)](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/garibaldie8.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie algebra structure*
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