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\begin{document}

\title{Erratum to "On $\delta$-derivations of Lie algebras and superalgebras"}
\author{Pasha Zusmanovich}
\address{
Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 
Tallinn 19086, Estonia
}
\email[as of April 19, 2024]{pasha.zusmanovich@gmail.com} 
\thanks{J. Algebra \textbf{410} (2014), 545--546}
\date{written July 15, 2013; last minor revision April 19, 2024}

\maketitle


In \cite{delta}, an incorrect definition of the Grassmann envelope
of a Lie superalgebra was used. Namely, if $L = L_0 \oplus L_1$ is a Lie
superalgebra, and $G = G_0 \oplus G_1$ is the Grassmann algebra, the 
multiplication on $G(L) = (L_0 \otimes G_0) \oplus (L_1 \otimes G_1)$
can be defined in two ways; as for ordinary algebras:
$$
[x \otimes g, y \otimes h] = [x,y] \otimes gh ,
$$
and as for superalgebras:
$$
[x \otimes g, y \otimes h] = (-1)^{\deg(g)\deg(y)} [x,y] \otimes gh ,
$$
where $x,y$ and $g,h$ are homogeneous elements of $L$ and $G$, respectively. 
These two multiplications lead to what is called in \cite[p.~72]{berezin}
the Grassmann envelopes of the first and second kind. However, as justly 
remarked in the footnote at the same page of \cite{berezin}, the first definition 
contradicts the Sign Rule, and will lead, sooner or later, to confusion. 
\cite{delta} is an instance of such a confusion:
it is easy to see that the Grassmann envelope of the first kind is, generally, 
not a Lie algebra, what defeats its main utility,
at least in the context of \cite{delta}.

To remedy this, one should adopt the definition of the Grassmann envelope of 
the second kind, and inject signs at appropriate places. First, the map in 
Lemma 4.1(ii) should be defined as
$$
a_0 \otimes b_0 + a_1 \otimes b_1 \mapsto 
D(a_0) \otimes \chi(b_0) + (-1)^{\deg(D)} D(a_1) \otimes \chi(b_1) .
$$

Second, the symmetric bilinear form used in the proof of Theorem 4.8
should be defined as
$$
(x \otimes g, x^\prime \otimes g^\prime) = 
(-1)^{\deg(g)\deg(x^\prime)} (x,x^\prime) f(gg^\prime) .
$$

No modifications in the proofs are needed.

Thanks are due to Liangyun Chen, Yao Ma, Yongzheng Zhang and Keli Zheng 
for bringing this defect to my attention, and to Dimitry Leites for getting
me acquainted with the 2nd edition of \cite{berezin}.



\begin{thebibliography}{B} 

\bibitem[B]{berezin} F. Berezin,  
\emph{Introduction to superanalysis}, 2nd ed., revised and edited by D. Leites,
with appendix \emph{Seminar on Supersymmetry. Vol. 1$\frac 12$}, 
MCCME, Moscow, 2013 (in Russian); 
D. Reidel, 1987 (English translation of the 1st ed.).

\bibitem[Z]{delta} P. Zusmanovich, 
\emph{On $\delta$-derivations of Lie algebras and superalgebras},
J. Algebra \textbf{324} (2010), 3470--3486; \textsf{arXiv:0907.2034}.

\end{thebibliography}

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