Bracket width of simple Lie algebras

TL;DR

This paper introduces the concept of bracket width in simple Lie algebras and provides the first examples of such algebras with bracket width greater than one, expanding understanding of algebraic complexity.

Contribution

It defines the bracket width of Lie algebras and presents the first known simple Lie algebras with bracket width exceeding one, specifically among polynomial vector fields.

Findings

Existence of simple Lie algebras with bracket width > 1

Examples found among polynomial vector fields on affine varieties

Extension of the concept of commutator width to Lie algebras

Abstract

The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of polynomial vector fields on smooth affine varieties.