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
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.