En (Lie algebra)

Dynkin diagrams

Finite
E3=A2A1	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-2.pngFile:Dyn2-node n3.png
E4=A4	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.png
E5=D5	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.png
E6	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.png
E7	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.png
E8	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.pngFile:Dyn2-3.pngFile:Dyn2-node n7.png
Affine (Extended)
E9 or E(1) 8 or E+ 8	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.pngFile:Dyn2-3.pngFile:Dyn2-node n7.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n8.png
Hyperbolic (Over-extended)
E10 or E(1)^ 8 or E++ 8	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.pngFile:Dyn2-3.pngFile:Dyn2-node n7.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n8.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n9.png
Lorentzian (Very-extended)
E11 or E+++ 8	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.pngFile:Dyn2-3.pngFile:Dyn2-node n7.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n8.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n9.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n10.png
Kac–Moody
E12 or E++++ 8	File:Dyn2-node n1.pngFile:Dyn2-3.pngFile:Dyn2-node n2.pngFile:Dyn2-3.pngFile:Dyn2-branch.pngFile:Dyn2-3.pngFile:Dyn2-node n4.pngFile:Dyn2-3.pngFile:Dyn2-node n5.pngFile:Dyn2-3.pngFile:Dyn2-node n6.pngFile:Dyn2-3.pngFile:Dyn2-node n7.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n8.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n9.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n10.pngFile:Dyn2-3.pngFile:Dyn2-nodeg n11.png
...

In mathematics, especially in Lie theory, E_n 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 E_n group is similar to the A_n 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 E_n is 9 − n.

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

  $\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& 0\\ 0& 0& 2\end{array}\right]$

• E₄ is another name for the Lie algebra A₄ of dimension 24, with Cartan determinant 5.

  $\left[\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right]$

• E₅ is another name for the Lie algebra D₅ of dimension 45, with Cartan determinant 4.

  $\left[\begin{array}{ccccc}2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 2& -1& -1\\ 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 2\end{array}\right]$

• E₆ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.

  $\left[\begin{array}{cccccc}2& -1& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0\\ 0& -1& 2& -1& 0& -1\\ 0& 0& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 0& 2\end{array}\right]$

• E₇ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.

  $\left[\begin{array}{ccccccc}2& -1& 0& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0& 0\\ 0& -1& 2& -1& 0& 0& -1\\ 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 0& 0& 2\end{array}\right]$

• E₈ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.

  $\left[\begin{array}{cccccccc}2& -1& 0& 0& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0& 0& 0\\ 0& -1& 2& -1& 0& 0& 0& -1\\ 0& 0& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 0& 0& 0& 2\end{array}\right]$

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.

  $\left[\begin{array}{ccccccccc}2& -1& 0& 0& 0& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0& 0& 0& 0\\ 0& -1& 2& -1& 0& 0& 0& 0& -1\\ 0& 0& -1& 2& -1& 0& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 2\end{array}\right]$

• 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 II_9,1 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:

  $\left[\begin{array}{cccccccccc}2& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 2& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 2& -1& 0& 0& 0& 0& 0& -1\\ 0& 0& -1& 2& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& -1& 2& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 2\end{array}\right]$

• 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.
• E_n for n ≥ 12 is a family of infinite-dimensional Kac–Moody algebras that are not well studied.

Root lattice

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

E_7+1⁄2

Main article: E7½

Landsberg and Manivel extended the definition of E_n 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 E_n series which was observed by Cvitanovic, Deligne, Cohen and de Man. E_7+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, 2_k1, 1_k2 polytopes based on E_n Lie algebras.

Further reading

• Class. Quantum Grav. 18 (2001) 4443-4460
• Guersey Memorial Conference Proceedings '94
• 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]