A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann

TL;DR

This paper identifies a flaw in a recent proof claiming that a finite-dimensional complex Lie algebra is semi-simple if and only if its Leibniz homology vanishes in positive dimensions, challenging a proposed characterization.

Contribution

It provides a counterexample to a conjecture linking semi-simplicity of Lie algebras with Leibniz homology vanishing, and points out an error in the proof by Burde and Wagemann.

Findings

The conjecture is false due to a counterexample.

The proof by Burde and Wagemann contains a mistake.

Leibniz homology does not characterize semi-simplicity as previously claimed.

Abstract

Our (weak) conjecture claims that a finite dimensional Lie algebra ${\bf g}$ over the field of complex numbers is semi-simple iff the Leibniz homology vanishes in positive dimensions $HL_i({\bf g})=0$, $i>0$. We will indicate a mistake in the recent proof of this conjecture due to Burde and Wagemann.