En (Lie algebra)

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

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

• 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

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.

E 7 + 1 ⁄ 2

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.

• 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 .

Further reading

• West, P. (2001). "E11 and M Theory". Classical and Quantum Gravity. 18 (21): 4443–4460. arXiv:. Bibcode:. doi:. S2CID . 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:. doi:. ISBN 978-3-540-59163-4. S2CID . Guersey Memorial Conference Proceedings '94
• Landsberg, J. M.; Manivel, L. (2006). . Advances in Mathematics. 201 (1): 143–179. arXiv:. doi:.
• 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