En (Lie algebra)
|E9 or E8(1) or E8+|
|E10 or E8(1)^ or E8++|
|E11 or E8+++|
|E12 or E8++++|
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.
- E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
- E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
- E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
- E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
- E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
Infinite-dimensional Lie algebras
- E9 is another name for the infinite-dimensional affine Lie algebra (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.
- 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:
- 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 an infinite-dimensional Kac–Moody algebra that has not been studied much.
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.
Landsberg and Manivel extended the definition of En for integer n to include the case n = 7½. 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. E7½ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.
- 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 Acad. Publ. pp. 109–128. MR 981374.
- West, P. (2001). "E11 and M Theory". Classical and Quantum Gravity. 18 (21): 4443–4460. arXiv:hep-th/0104081. doi:10.1088/0264-9381/18/21/305. Class.Quant.Grav. 18 (2001) 4443-4460
- Gebert, R. W.; Nicolai, H. (1994). "E10 for beginners". arXiv:hep-th/9411188 [hep-th]. Guersey Memorial Conference Proceedings '94
- Landsberg, J. M. Manivel, L. The sextonions and E7½. Adv. Math. 201 (2006), no. 1, 143-179.
- Connections between Kac-Moody algebras and M-theory, Paul P. Cook, 2006
- A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002