Gibbs isotherm

Dynkin diagrams

Finite
E3=A2A1	Template:Dynkin2
E4=A4	Template:Dynkin2
E5=D5	Template:Dynkin2
E6	Template:Dynkin2
E7	Template:Dynkin2
E8	Template:Dynkin2
Affine (Extended)
E9 or E8(1) or E8+	Template:Dynkin2
Hyperbolic (Over-extended)
E10 or E8(1)^ or E8++	Template:Dynkin2
Lorentzian (Very-extended)
E11 or E8+++	Template:Dynkin2
Kac–Moody
E12 or E8++++	Template:Dynkin2
...

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 ${\tilde{E}}_8$ (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.

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

• E₁₀ (or 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 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 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.
• E_n for n≥12 is an infinite dimensional Kac–Moody algebra that has not been studied much.

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.

E7½

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Landsberg and Manivel extended the definition of E_n 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 E_n series which was observed by Cvitanovic, Deligne, Cohen and de Man. E_7½ 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.

References

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• Template:Cite arxiv Guersey Memorial Conference Proceedings '94
• Landsberg, J. M. Manivel, L. The sextonions and E_7½. 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]