A new construction of ${\tilde{D}}_5$-singularities and generalization of Slodowy slices

Abstract

Any simple elliptic singularity of type ${\tilde{D}}_5$ can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional "good slices" in the semi-simple Lie algebra $\mathfrak{s}l(2,\mathbb{C})\oplus \mathfrak{s}l(2,\mathbb{C})$. We describe these new slices purely by the structure of the Lie algebra. We also construct the semi-universal deformation spaces of ${\tilde{D}}_5$-singularities by using the 4-dimensional "good slices".

19 pages