In mathematics, especially in Lie theory, En 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, E2 and E4 are used as names for G2 and F4.
The determinant of the Cartan matrix for En is 9 − n. 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,...,1|3) of norm n × 12 − 32 = n − 9.
Landsberg and Manivel extended the definition of En for integer n to include the case n = 7+1⁄2.
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.