A new construction of -singularities and generalization of Slodowy slices

Any simple elliptic singularity of type can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional “good slices” in the semi-simple Lie algebra sl(2,C)⊕sl(2,C). We describe these new slices purely by the structure of the Lie algebra. We also construct the semi-universal deformation spaces of -singularities by using the 4-dimensional “good slices”.