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\title[Mock-Lie algebras]{Special and exceptional mock-Lie algebras \\ (extended abstract)}
%----------Author1
\author{Pasha Zusmanovich}
\address{%
Department of Mathematics \\
University of Ostrava \\
Ostrava, Czech Republic
}
\email{pasha.zusmanovich@osu.cz}
\begin{document}
\maketitle
%\section*{Introduction}
Mock-Lie algebras are algebras satisfying two identities: commutativity
$$
xy = yx
$$
and the Jacobi identity
$$
(xy)z + (zx)y + (yz)x = 0 .
$$
It is almost immediate that such algebras are Jordan algebras of nil index
$3$ (i.e., $x^3 = 0$). Conversely, over a field of characteristic $\ne 2,3$,
any Jordan algebra of nil index $3$ is mock-Lie. Such algebras appeared in the
literature under different names: Lie-Jordan, Jacobi-Jordan,
``commutative'' Lie algebras, etc. They live a dual life: as a very particular
class of Jordan algebras, and as strange cousins of Lie algebras, and this work
is an interesting blend of Lie and Jordan theories (and, of course,
computer algebra).
The main question we are concerned with is which of those algebras admit a
faithful representation, or, what is the same, admit embedding into an
associative algebra (i.e., in Lie parlance, satisfy the Ado theorem, or, in
Jordan parlance, are special).
The arguments used to establish the Ado theorem and the
Poincar\'e--Birkhoff--Witt theorem -- a fact closely related with the
possibility of embedding of Lie algebras into associative ones -- fail
in somewhat curios ways in the mock-Lie case (one of those ways involves
calculation of Gr\"obner bases of universal enveloping algebras of mock-Lie algebras).
Mock-Lie algebras of low dimension ($\le 6$ at least) are special (i.e.,
embedded into an associative algebra). On the other hand, an exceptional (i.e.,
not special) mock-Lie algebra was constructed a long time ago in an unpublished
preprint \cite{hjs}, and we reproduce these old efforts, showing how for any Jordan s-identity
(Glennie, Thedy, Medvedev, etc.) one may produce a mock-Lie algebra which does
not satisfy this identity, and hence is exceptional. The minimal dimension of
so constructed algebra is $44$.
Substantial computer calculations are involved, utilizing
Albert \cite{albert-mod} and GAP.
There are many open questions related to mock-Lie algebras: about minimal
dimension and minimal degree of nilpotency of exceptional algebras, about
cohomology theory, about mock-Lie and dual to it operads, etc.
Based on \cite{mock-lie}.
\begin{thebibliography}{2}
\bibitem{hjs}
I.R. Hentzel, D.P. Jacobs, and S.R. Sverchkov,
\emph{On exceptional nil of index $3$ Jordan algebras}, Preprint,
Novosibirsk State Univ., 1997.
\bibitem{mock-lie}
P. Zusmanovich, \emph{Special and exceptional mock-Lie algebras},
Lin. Algebra Appl. \textbf{518} (2017), 79--96;
arXiv:1608.05861.
\bibitem{albert-mod} Albert;
\url{http://www1.osu.cz/~zusmanovich/albert/}
\end{thebibliography}
\end{document}