En (Lie algebra)

Finite-dimensional Lie algebras

The En group is similar to the An 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 En is 9 − n.

• 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 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 II9,1 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.
• En for n ≥ 12 is a family of infinite-dimensional Kac–Moody algebras that are not well studied.

Root lattice

The root lattice of En has determinant 9 − n, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Zn,1 that are orthogonal to the vector (1,1,1,1,...,13) of norm n × 12 − 32 = n − 9.

E_{{{frac|7|1|2}}}

Main article: E7½

Landsberg and Manivel extended the definition of En for integer n to include the case n = . They did this in order to fill the "hole" in dimension formulae for representations of the En 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.

References