En (Lie algebra)

Dynkin diagrams  Finite

E3=A2A1
E4=A4
E5=D5

Affine (Extended)
E9 or E8(1) or E8+
Hyperbolic (Over-extended)
E10 or E8(1)^ or E8++
Lorentzian (Very-extended)
E11 or E8+++
Kac–Moody
E12 or E8++++
...

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.

## Contents

* 1 Finite-dimensional Lie algebras
* 2 Infinite-dimensional Lie algebras
* 3 Root lattice
* 4 E7½
* 5 See also
* 6 References
* 7 Further reading

## Finite-dimensional Lie algebras

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.

[2−10−120002]
* E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.

[2−100−12−100−12−100−12]
* E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.

[2−1000−12−1000−12−1−100−12000−102]
* E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.

[2−10000−12−10000−12−10−100−12−10000−12000−1002]
* E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.

[2−100000−12−100000−12−100−100−12−100000−12−100000−12000−10002]
* E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.

[2−1000000−12−1000000−12−1000−100−12−1000000−12−1000000−12−1000000−12000−100002]

## Infinite-dimensional Lie algebras

* E9 is another name for the infinite-dimensional affine Lie algebra E~8 (also as E8+ or E8(1) 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.

[2−10000000−12−10000000−12−10000−100−12−10000000−12−10000000−12−10000000−12−10000000−12000−1000002]
* E10 (or E8++ or E8(1)^ 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:

[2−100000000−12−100000000−12−100000−100−12−100000000−12−100000000−12−100000000−12−100000000−12−100000000−12000−10000002]
* E11 (or E8+++ 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

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.

## E7½

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

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

## References

* 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.. 250. Dordrecht: Kluwer Academic Publishers Group. pp. 109–128.

## Further reading

* West, P. (2001). "E11 and M Theory". Classical and Quantum Gravity 18 (21): 4443–4460. doi:10.1088/0264-9381/18/21/305. Bibcode: 2001CQGra..18.4443W. Class. Quantum Grav. 18 (2001) 4443-4460
* Gebert, R. W.; Nicolai, H. (1994). "E 10 for beginners". E10 for beginners. Lecture Notes in Physics. 447. pp. 197–210. doi:10.1007/3-540-59163-X_269. ISBN 978-3-540-59163-4. Guersey Memorial Conference Proceedings '94
* Landsberg, J. M.; Manivel, L. (2006). "The sextonions and E7½". Advances in Mathematics 201 (1): 143–179. 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]

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