By compactifying with a root lattice in the case of A-D-E lattice one obtains the affine Lie algebra of the corresponding Lie group at level 1. That is basically the Frenkel-Kac construction:
I. B. Frenkel and V. G. Kac. Basic representations of affine Lie algebras and dual resonance models. Invent. Math., 62(1):23–66, 1980.