  Using tight binding model, lattice QFT and group theory methods, we study a
class of lattice QFT models that are cousins of graphene; and which are
classified by finite dimensional ADE Lie groups containing the usual
crystallographic symmetries as discrete subgroups. We show in particular that
the electronic properties of the 1D lattice poly-acetylene chain are given by a
SU(2) model and those of the well known 2D graphene by SU(3). We also give two
other models classified by SU(4) and SO(6) symmetries; they respectively
describe 3D diamond and 3D lattice with octahedral sites. It is shown as well
that the dispersion energies of this set of models are completely characterized
by the roots of the Lie algebras underlying the symmetry groups. Other
features, such as SO(5) lattice involving sp^{3}d hybridization as well as the
relation between the 4D hyperdiamond, having a SU(5) symmetry and the 4D
lattice QCD, are also discussed.
