\titleofpresentation{Algebras with skew-symmetric identity of degree
3}
\presenter{A. S. Dzhumadil'daev} 
 
Algebras with one of the following identities are considered:
$$[[t_1,t_2],t_3]+[[t_2,t_3],t_1]+[[t_3,t_1],t_2]=0,
\mbox{(Lie-Admissible)}$$
$$[t_1,t_2]t_3+[t_2,t_3]t_1+[t_3,t_1]t_2=0, \mbox{
($0$-Lie-Admissible (shortly $0$-Alia))}$$
$$\{[t_1,t_2],t_3\}+\{[t_2,t_3],t_1\}+\{[t_3,t_1],t_2\}=0,
\mbox{($1$-Lie-admissible (shortly $1$-Alia))}$$ where
$[t_1,t_2]=t_1t_2-t_2t_1$ and $\{t_1,t_2\}=t_1t_2+t_2t_1.$ For
algebra $A=(A,\circ)$ with multiplication $\circ$ denote by
$A^{(q)}$ an algebra with vector space $A$ and multiplication
$a\circ_q b=a\circ b+q\,b\circ a.$
 
\begin{theorem*} Any algebra with a skew-symmetric identity of
degree $3$ is (anti-) isomorphic to one of the following algebras:
\begin{itemize}
\item Lie-admissible algebra
\item $0$-Alia algebra
\item $1$-Alia algebra
\item algebra of the form $A^{(q)}$ for some left-Alia algebra $A$
and $q\in K$, such that $q^2\ne 0,1.$
\end{itemize}
\end{theorem*}

\begin{example}
$({\bf C}[x],\circ)$ under multiplication $a\circ
b=\partial(a)\partial^2(b)$ is $1$-Alia and simple.
\end{example}

\begin{example}
$({\bf C}[x],\star),$ where $a\star
b=\partial^3(a)b+4\partial^2(a)\partial(b)+5\partial(a)\partial^2(b)+2
a\,\partial^3(b),$ is $0$-Alia and simple.
\end{example}
 
\begin{ourbiblio}{99} 
 
\bibitem{A. Dzhumadil'daev}
{\em $N$-commutators,} Comm.Math. Helvetici, {\bf 79}(2004),
No.3, p. 516-553. Preprint available math.RA/0203036
\end{ourbiblio}
\presenteraddress{S. Demirel University, Almaty}  

\email{askar@math.kz}

