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, E₂ and E₄ are used as names for G₂ and F₄.

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.

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.

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. E_7½ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

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