Singular deformations of Lie algebras on an example

TL;DR

This paper introduces the concept of singular one-parameter deformations of Lie algebras and demonstrates such a deformation in an infinite-dimensional algebra of polynomial vector fields.

Contribution

It defines singular deformations of Lie algebras and provides a concrete example involving polynomial vector fields with trivial 1-jet.

Findings

Existence of singular deformation in an infinite-dimensional Lie algebra

Introduction of the notion of singular one-parameter deformation

Application to polynomial vector fields in C

Abstract

The notion of singular one-parameter deformation of a Lie algebra is introduced. It is shown that the complex infinite-dimensional Lie algebra of polynomial vector fields in C with trivial 1-jet at the origin has such singular deformation.