En (Lie Algebra) - Infinite Dimensional Lie Algebras

Infinite Dimensional Lie Algebras

E₉ is another name for the infinite dimensional affine Lie algebra (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.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$
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:
- $\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$
E₁₁ (or E₈+++ as a (three-node) very-extended E₈) is a Lorentzian algebra, containining 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.

Read more about this topic: En (Lie Algebra)

Famous quotes containing the words infinite, dimensional and/or lie:

“The universe is then one, infinite, immobile.... It is not capable of comprehension and therefore is endless and limitless, and to that extent infinite and indeterminable, and consequently immobile.”
—Giordano Bruno (1548–1600)

“I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.”
—Kristin Hunter (b. 1931)

“I will name you the degrees. The first, the Retort Courteous; the second, the Quip Modest; the third, the Reply Churlish; the fourth, the Reproof Valiant; the fifth, the Countercheck Quarrelsome; the sixth, the Lie with Circumstance; the seventh, the Lie Direct.”
—William Shakespeare (1564–1616)