Deformations of pairs of Kleinian singularities

TL;DR

This paper classifies deformations of algebraic invariants arising from Kleinian group inclusions, extending previous classifications to cases where a subgroup is normal, with implications for algebraic geometry and singularity theory.

Contribution

It provides a classification of deformations of inclusions of Kleinian group invariants when the smaller subgroup is normal, expanding the understanding of their deformation theory.

Findings

Classified deformations of invariants for normal subgroup inclusions

Extended previous deformation classifications to new subgroup cases

Linked deformations to algebraic and geometric properties of singularities

Abstract

Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger.