E_n (Lie algebra)

Dynkin diagrams
Finite
E₃=A₂A₁
E₄=A₄
E₅=D₅
E₆
E₇
E₈
Affine (Extended)
E₉ or E⁽¹⁾ ₈ or E⁺ ₈
Hyperbolic (Over-extended)
E₁₀ or E^{(1)^} ₈ or E⁺⁺ ₈
Lorentzian (Very-extended)
E₁₁ or E⁺⁺⁺ ₈
Kac–Moody
E₁₂ or E⁺⁺⁺⁺ ₈
...

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

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

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.

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. Vol. 250. Dordrecht: Kluwer Academic Publishers Group. pp. 109–128. MR 0981374.