En (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 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 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.

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

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

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

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

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

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

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

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

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

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

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

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.

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

• 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:

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

• 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.

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Main article: E7½

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.

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. Vol. 250. Dordrecht: Kluwer Academic Publishers Group. 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. S2CID 250872099. Class. Quantum Grav. 18 (2001) 4443-4460
• Gebert, R. W.; Nicolai, H. (1994). "E 10 for beginners". E₁₀ for beginners. Lecture Notes in Physics. Vol. 447. pp. 197–210. arXiv:hep-th/9411188. doi:10.1007/3-540-59163-X_269. ISBN 978-3-540-59163-4. S2CID 14570784. Guersey Memorial Conference Proceedings '94
• Landsberg, J. M.; Manivel, L. (2006). "The sextonions and E_7½". Advances in Mathematics. 201 (1): 143–179. arXiv:math.RT/0402157. doi:10.1016/j.aim.2005.02.001.
• 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