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 .

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.
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• E₄ is another name for the Lie algebra A₄ of dimension 24, with Cartan determinant 5.
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• E₅ is another name for the Lie algebra D₅ of dimension 45, with Cartan determinant 4.
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• E₆ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
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• E₇ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
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• E₈ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
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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.
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• 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:
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• 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

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

Further reading

• Class. Quantum Grav. 18 (2001) 4443-4460
• Guersey Memorial Conference Proceedings '94
• 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