En (Lie algebra)

Finite-dimensional Lie algebras

• E₃ is another name for the Lie algebra A₁A₂ of dimension 11, with Cartan determinant 6.
• :
• E₄ is another name for the Lie algebra A₄ of dimension 24, with Cartan determinant 5.
• :
• E₅ is another name for the Lie algebra D₅ of dimension 45, with Cartan determinant 4.
• :
• E₆ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
• :
• E₇ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
• :
• E₈ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
• :

  Infinite-dimensional Lie algebras

• E₉ is another name for the infinite-dimensional affine Lie algebra as a corresponding to the Lie algebra of type E₈. E₉ has a Cartan matrix with determinant 0.
• :
• E₁₀ ^ as a 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:
• :
• E₁₁ 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

E7½