En (Lie algebra)

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Dynkin diagrams
Finite
E3=A2A1 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-2.pngDyn2-node n3.png
E4=A4 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.png
E5=D5 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.png
E6 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.png
E7 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.png
E8 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.pngDyn2-3.pngDyn2-node n7.png
Affine (Extended)
E9 or E8(1) or E8+ Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.pngDyn2-3.pngDyn2-node n7.pngDyn2-3.pngDyn2-nodeg n8.png
Hyperbolic (Over-extended)
E10 or E8(1)^ or E8++ Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.pngDyn2-3.pngDyn2-node n7.pngDyn2-3.pngDyn2-nodeg n8.pngDyn2-3.pngDyn2-nodeg n9.png
Lorentzian (Very-extended)
E11 or E8+++ Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.pngDyn2-3.pngDyn2-node n7.pngDyn2-3.pngDyn2-nodeg n8.pngDyn2-3.pngDyn2-nodeg n9.pngDyn2-3.pngDyn2-nodeg n10.png
Kac–Moody
E12 or E8++++ Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node n4.pngDyn2-3.pngDyn2-node n5.pngDyn2-3.pngDyn2-node n6.pngDyn2-3.pngDyn2-node n7.pngDyn2-3.pngDyn2-nodeg n8.pngDyn2-3.pngDyn2-nodeg n9.pngDyn2-3.pngDyn2-nodeg n10.pngDyn2-3.pngDyn2-nodeg n11.png
...

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[edit]

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.
    \left [
\begin{smallmatrix}
 2 & -1 &  0 \\
-1 &  2 &  0 \\
 0 &  0 &  2 
\end{smallmatrix}\right ]
  • E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
    \left [
\begin{smallmatrix}
 2 & -1 &  0 &  0 \\
-1 &  2 & -1&  0  \\
 0 & -1 &  2 & -1 \\
 0 &  0 & -1 &  2 
\end{smallmatrix}\right ]
  • E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
    \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]
  • E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
    \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]
  • E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
    \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]
  • E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
    \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]

Infinite-dimensional Lie algebras[edit]

  • E9 is another name for the infinite-dimensional affine Lie algebra {\tilde{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.
    \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]
  • 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:
    • \left [
\begin{smallmatrix}
 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{smallmatrix}\right ]
  • E11 (or E8+++ as a (three-node) very-extended E8) is a Lorentzian algebra, containining 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.

Root lattice[edit]

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[edit]

Main article: E7½

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. E has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

See also[edit]

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

References[edit]

  • 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. 

Further reading[edit]

  • 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 E. Adv. Math. 201 (2006), no. 1, 143-179.
  • 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]