Wide simple Lie algebras

We say that a group G is wide if it contains an element which is not representable 

as a single commutator of elements of G. Recently it was proven that a finite simple 

group cannot be wide, thus confirming a conjecture of Ore of 1950's. On the other hand, 

during the past decades there were discovered several examples of wide infinite simple 

groups. 

In a similar vein, we say that a Lie algebra is wide if it contains an element which is not 

representable as a single Lie bracket. A natural question to ask is whether there exist 

wide simple Lie algebras. Our goal is to present first examples of such Lie algebras.

The simplest example relies on a recent work of Billig and Futorny on Lie algebras of vector 

fields on smooth affine varieties.

This talk is based on a work in progress, joint with Andriy Regeta.