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

TL;DR

This paper introduces a novel Lie algebra-based construction of D_5-singularities using special slices, and extends the understanding of their semi-universal deformations through these slices.

Contribution

It provides a new Lie algebraic approach to construct D_5-singularities and generalizes Slodowy slices for this purpose.

Findings

Constructed D_5-singularities via intersection with good slices.

Described slices purely through Lie algebra structure.

Built semi-universal deformation spaces using these 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 ${\frak sl}(2, {\mathbb C}) \oplus {\frak sl}(2, {\mathbb C})$. 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".