E_n (Lie algebra)

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.

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 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 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.

See also

• k₂₁, 2_k1, 1_k2 polytopes based on E_n Lie algebras.

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 0981374.

Further reading

• West, P. (2001). "E₁₁ and M Theory". Classical and Quantum Gravity. 18 (21): 4443–4460. arXiv:hep-th/0104081. Bibcode:2001CQGra..18.4443W. doi:10.1088/0264-9381/18/21/305. Class. Quantum Grav. 18 (2001) 4443-4460
• Gebert, R. W.; Nicolai, H. (1994). "E₁₀ for beginners". Lecture Notes in Physics: 197–210. arXiv:hep-th/9411188. doi:10.1007/3-540-59163-X_269. 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
• A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002