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\begin{document}

\title{Erratum to \\ ``A converse to the Second Whitehead Lemma''}
\author{Pasha Zusmanovich}
\address{
Institute of Mathematics and Statistics, University of S\~{a}o Paulo,
Brazil
}
\email[as of April 20, 2024]{pasha.zusmanovich@gmail.com}
\date{Written April 27, 2014; last minor revision April 20, 2024}
\thanks{J. Lie Theory \textbf{24} (2014), no.4, 1207--1208}

\maketitle

In \cite{z1}, a result from \cite{rozen} is used, claiming that the 
Hochschild--Serre spectral sequence abutting to the cohomology of the 
semidirect sum $L = S + I$ of Lie algebras $S$ and $I$  
($S$ is a subalgebra, $I$ is an ideal, i.e. $S$ acts on $I$) with coefficients
in an arbitrary $L$-module $V$, with respect to the ideal $I$, stabilizes at the 
$E_2$ page (\cite[Lemma 1]{rozen} and \cite[Proposition 1.3]{z1}). 
In the whole generality, this statement is false. In fact, as shown in 
\cite{barnes}, this spectral sequence can stabilize at arbitrarily large step.

This statement is true, however, in many special cases: for example,
if the base field is of characteristic zero, $S$ is a 
finite-dimensional semisimple Lie algebra, and $V$ is finite-dimensional 
(\cite[Theorem 13]{hs}). 
More sufficient conditions guaranteeing stabilization at $E_2$ can be found in 
\cite[Theorem 3]{barnes}, \cite[Corollaries 1.5, 1.6 and Theorem 1.7]{dp1}, 
and \cite[Theorem 1.2]{dp2}.

The main result (Theorem 0.2) of \cite{z1} can be salvaged in the following way.
The above-mentioned general erroneous claim was used once, in the proof of 
Lemma 2.1. Lemma 2.1, in its turn, was used in the proof of Theorem 0.2 twice: 
first time (in the proof of Lemma 2.4) in the situation where $S$ is semisimple,
what is covered by the Hochschild--Serre result mentioned above,
and the second time in the situation where $L = S + I$ is the direct sum of 
algebras. In the latter case, one can invoke the K\"unneth theorem instead of 
Proposition 1.3.

I am grateful to Donald Barnes who pointed out this error.


\begin{thebibliography}{DP2} 

\bibitem[B]{barnes} D.W. Barnes, 
\emph{On the length of the spectral sequence of a Lie algebra extension},
Proc. Amer. Math. Soc. \textbf{129} (2001), 347--350.

\bibitem[DP1]{dp1} D. Degrijse and N. Petrosyan, 
\emph{Characteristic classes for cohomology of split Hopf algebra extensions},
J. Algebra \textbf{332} (2011), 366--385; \textsf{arXiv:1002.0098}.

\bibitem[DP2]{dp2} \bysame{} and \bysame, 
\emph{On cohomology of split Lie algebra extensions},
J. Lie Theory 22 (2012), 1--15; \textsf{arXiv:0911.0545}.

\bibitem[HS]{hs} G. Hochschild and J.-P. Serre, 
\emph{Cohomology of Lie algebras}, Ann. Math. \textbf{57} (1953), 591--603.

\bibitem[R]{rozen} B.I. Rozenfeld, 
\emph{Cohomology of the algebra of formal universal differential operators},
Funkts. Anal. Prilozh. \textbf{9} (1975), $\mathcal{N}$2, 42--47 (in Russian);
Funct. Anal. Appl. \textbf{9} (1975), 126--130 (English translation).

\bibitem[Z]{z1} P. Zusmanovich, \emph{A converse to the Second Whitehead Lemma},
J. Lie Theory \textbf{18} (2008), 295--299; \textsf{arXiv:0704.3864}.

\end{thebibliography}

\end{document}
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