En (Lie algebra)

Dynkin diagrams  Finite

E3=A2A1 |
E4=A4 |
E5=D5 |
E6 |
E7 |
E8 |
Affine (Extended)
E9 or E(1)
8 or E+
8 |
Hyperbolic (Over-extended)
E10 or E(1)^
8 or E++
8 |
Lorentzian (Very-extended)
E11 or E+++
8 |
Kac–Moody
E12 or E++++
8 |
...

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and k, with k = n − 4.

In some older books and papers, E2 and E4 are used as names for G2 and F4.

## Finite-dimensional Lie algebras

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The En group is similar to the An group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, −1 above and below the diagonal, except for the last row and column, have −1 in the third row and column. The determinant of the Cartan matrix for En is 9 − n.

* E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6.

[ 2 − 1 0 − 1 2 0 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0\\\\-1&2&0\\\0&0&2\end{matrix}}\right]}
* E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.

[ 2 − 1 0 0 − 1 2 − 1 0 0 − 1 2 − 1 0 0 − 1 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0\\\\-1&2&-1&0\\\0&-1&2&-1\\\0&0&-1&2\end{matrix}}\right]}
* E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.

[ 2 − 1 0 0 0 − 1 2 − 1 0 0 0 − 1 2 − 1 − 1 0 0 − 1 2 0 0 0 − 1 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0\\\\-1&2&-1&0&0\\\0&-1&2&-1&-1\\\0&0&-1&2&0\\\0&0&-1&0&2\end{matrix}}\right]}
* E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.

[ 2 − 1 0 0 0 0 − 1 2 − 1 0 0 0 0 − 1 2 − 1 0 − 1 0 0 − 1 2 − 1 0 0 0 0 − 1 2 0 0 0 − 1 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0&0\\\\-1&2&-1&0&0&0\\\0&-1&2&-1&0&-1\\\0&0&-1&2&-1&0\\\0&0&0&-1&2&0\\\0&0&-1&0&0&2\end{matrix}}\right]}
* E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.

[ 2 − 1 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 − 1 2 − 1 0 0 − 1 0 0 − 1 2 − 1 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 − 1 2 0 0 0 − 1 0 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0&0&0\\\\-1&2&-1&0&0&0&0\\\0&-1&2&-1&0&0&-1\\\0&0&-1&2&-1&0&0\\\0&0&0&-1&2&-1&0\\\0&0&0&0&-1&2&0\\\0&0&-1&0&0&0&2\end{matrix}}\right]}
* E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.

[ 2 − 1 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 − 1 2 − 1 0 0 0 − 1 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 0 0 0 − 1 0 0 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0&0&0&0\\\\-1&2&-1&0&0&0&0&0\\\0&-1&2&-1&0&0&0&-1\\\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&0\\\0&0&-1&0&0&0&0&2\end{matrix}}\right]}

## Infinite-dimensional Lie algebras

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* E9 is another name for the infinite-dimensional affine Lie algebra Ẽ8 (also as E+
8 or E(1)
8 as a (one-node) extended E8) (or E8 lattice) corresponding to the Lie algebra of type E8. E9 has a Cartan matrix with determinant 0.

[ 2 − 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 − 1 0 0 − 1 2 − 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 − 1 2 0 0 0 − 1 0 0 0 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0&0&0&0&0\\\\-1&2&-1&0&0&0&0&0&0\\\0&-1&2&-1&0&0&0&0&-1\\\0&0&-1&2&-1&0&0&0&0\\\0&0&0&-1&2&-1&0&0&0\\\0&0&0&0&-1&2&-1&0&0\\\0&0&0&0&0&-1&2&-1&0\\\0&0&0&0&0&0&-1&2&0\\\0&0&-1&0&0&0&0&0&2\end{matrix}}\right]}
* E10 (or E++
8 or E(1)^
8 as a (two-node) over-extended E8) is an infinite-dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II9,1 of dimension 10. Some of its root multiplicities have been calculated; for small roots the multiplicities seem to be well behaved, but for larger roots the observed patterns break down. E10 has a Cartan matrix with determinant −1:

[ 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 − 1 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 0 0 0 0 − 1 2 0 0 0 − 1 0 0 0 0 0 0 2 ] {\displaystyle \left[{\begin{matrix}2&-1&0&0&0&0&0&0&0&0\\\\-1&2&-1&0&0&0&0&0&0&0\\\0&-1&2&-1&0&0&0&0&0&-1\\\0&0&-1&2&-1&0&0&0&0&0\\\0&0&0&-1&2&-1&0&0&0&0\\\0&0&0&0&-1&2&-1&0&0&0\\\0&0&0&0&0&-1&2&-1&0&0\\\0&0&0&0&0&0&-1&2&-1&0\\\0&0&0&0&0&0&0&-1&2&0\\\0&0&-1&0&0&0&0&0&0&2\end{matrix}}\right]}
* E11 (or E+++
8 as a (three-node) very-extended E8) is a Lorentzian algebra, containing one time-like imaginary dimension, that has been conjectured to generate the symmetry "group" of M-theory.
* En for n ≥ 12 is a family of infinite-dimensional Kac–Moody algebras that are not well studied.

## Root lattice

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The root lattice of En has determinant 9 − n, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Zn,1 that are orthogonal to the vector (1,1,1,1,...,1|3) of norm n × 12 − 32 = n − 9.

## E7+1⁄2

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Main article: E7½

Landsberg and Manivel extended the definition of En for integer n to include the case n = 7+1⁄2. They did this in order to fill the "hole" in dimension formulae for representations of the En series which was observed by Cvitanovic, Deligne, Cohen and de Man. E7+1⁄2 has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

## See also

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* k21, 2k1, 1k2 polytopes based on En Lie algebras.

## References

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* Kac, Victor G; Moody, R. V.; Wakimoto, M. (1988). "On E10". Differential geometrical methods in theoretical physics (Como, 1987). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 250\. Dordrecht: Kluwer Academic Publishers Group. pp. 109–128\. MR 0981374.

## Further reading

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* West, P. (2001). "E11 and M Theory". Classical and Quantum Gravity. 18 (21): 4443–4460\. arXiv:hep-th/0104081. Bibcode:2001CQGra..18.4443W. doi:10.1088/0264-9381/18/21/305. S2CID 250872099. Class. Quantum Grav. 18 (2001) 4443-4460
* Gebert, R. W.; Nicolai, H. (1994). "E 10 for beginners". E10 for beginners. Lecture Notes in Physics. Vol. 447\. pp. 197–210\. arXiv:hep-th/9411188. doi:10.1007/3-540-59163-X_269. ISBN 978-3-540-59163-4. S2CID 14570784. Guersey Memorial Conference Proceedings '94
* Landsberg, J. M.; Manivel, L. (2006). "The sextonions and E7½". Advances in Mathematics. 201 (1): 143–179\. arXiv:math.RT/0402157. doi:10.1016/j.aim.2005.02.001.
* Connections between Kac-Moody algebras and M-theory, Paul P. Cook, 2006 [1]
* A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002 [2]