En (Lie algebra)

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

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

Main article: E7½

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.

References

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.

En (Lie algebra)

Finite-dimensional Lie algebras

Infinite-dimensional Lie algebras

Root lattice

E7½

See also

References

Further reading