Dynkin diagrams

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

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.

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0\\-1&2&0\\0&0&2\end{smallmatrix))\right]}$

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

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix))\right]}$

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

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&-1\\0&0&-1&2&0\\0&0&-1&0&2\end{smallmatrix))\right]}$

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

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&-1\\0&0&-1&2&-1&0\\0&0&0&-1&2&0\\0&0&-1&0&0&2\end{smallmatrix))\right]}$

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

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&-1\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&0\\0&0&-1&0&0&0&2\end{smallmatrix))\right]}$

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

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&-1\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&2\end{smallmatrix))\right]}$

Infinite-dimensional Lie algebras

• E₉ is another name for the infinite-dimensional affine Lie algebra ${\displaystyle {\tilde {E))_{8))$ (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.

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0\\0&0&0&-1&2&-1&0&0&0\\0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&2\end{smallmatrix))\right]}$

• 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 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:

  ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0&0\\0&0&0&-1&2&-1&0&0&0&0\\0&0&0&0&-1&2&-1&0&0&0\\0&0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&0&2\end{smallmatrix))\right]}$

• E₁₁ (or E₈⁺⁺⁺ as a (three-node) very-extended E₈) is a Lorentzian algebra, 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.

E7½

Landsberg and Manivel extended the definition of E_n for integer n to include the case n = 71⁄2. 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_71⁄2 has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.