En (Lie algebra) explained

Dynkin diagrams

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

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 .

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 .

• 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 or E 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 or E 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 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 is a family of infinite-dimensional Kac–Moody algebras that are not well studied.

Root lattice

The root lattice of E_n has determinant, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Z_n,1 that are orthogonal to the vector of norm = .

E

See main article: E7½.

Landsberg and Manivel extended the definition of E_n for integer n to include the case n = . 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 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

• Book: Kac . Victor G . Moody . R. V. . Wakimoto . M. . Differential geometrical methods in theoretical physics (Como, 1987) . Kluwer Academic Publishers Group . Dordrecht . NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. . 981374 . 1988 . 250 . On E₁₀ . 109–128 .

Further reading

• E₁₁ and M Theory . 2001 . West . P. . 10.1088/0264-9381/18/21/305 . Classical and Quantum Gravity . 18 . 21 . 4443–4460 . hep-th/0104081. 2001CQGra..18.4443W . 250872099 . Class. Quantum Grav. 18 (2001) 4443-4460
• Book: E₁₀ for beginners . 1994 . hep-th/9411188 . Gebert . R. W. . Nicolai . H. . E 10 for beginners . 10.1007/3-540-59163-X_269 . Lecture Notes in Physics . 447 . 197–210. 978-3-540-59163-4 . 14570784 . Guersey Memorial Conference Proceedings '94
• J. M.. Landsberg. L.. Manivel. The sextonions and E_7½. Advances in Mathematics. 2006. 201. 1. 143–179. math.RT/0402157. 10.1016/j.aim.2005.02.001. free.
• Connections between Kac-Moody algebras and M-theory, Paul P. Cook, 2006 https://arxiv.org/abs/0711.3498
• A class of Lorentzian Kac-Moody algebras, Matthias R. Gaberdiel, David I. Olive and Peter C. West, 2002 https://arxiv.org/abs/hep-th/0205068