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 "k" = "n" − 4.
In some older books and papers, "E"2 and "E"4 are used as names for "G"2 and "F"4.
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 9 − "n".
Root lattice.
The root lattice of E"n" has determinant 9 − "n", and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Z"n",1 that are orthogonal to the vector (1,1,1,1...,1|3) of norm "n" × 12 − 32 = "n" − 9.
E7½.
Landsberg and Manivel extended the definition of E"n" for integer "n" to include the case "n" = 7½. 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. E7½ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.