A six-dimensional semidirect product example shows Proposition 3.1 of Burde-Wagemann (arXiv:1908.05963) is false, invalidating their proposed proof of the weak Leibniz homology conjecture.
Sympathetic Lie algebras and adjoint cohomology for Lie algebras
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abstract
We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.
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A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann
A six-dimensional semidirect product example shows Proposition 3.1 of Burde-Wagemann (arXiv:1908.05963) is false, invalidating their proposed proof of the weak Leibniz homology conjecture.