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\title{A commutative $2$-cocycles approach to classification of simple Novikov algebras}
\author{Pasha Zusmanovich \\ (joint work in progress with Askar Dzhumadil'daev)}
\institute{Tallinn University of Technology}
\date{July 29, 2011}
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\begin{block}{What is a Novikov algebra?}
\begin{enumerate}[(i)]
\item
Left-symmetric (aka Vinberg, pre-Lie, chronological) identity:
$$
x(yz) - (xy)z = y(xz) - (yx)z
$$
\item
$$
(xy)z = (xz)y
$$
Equivalently:
\begin{align*}
[L_x,L_y] &= L_{[x,y]} \\
[R_x,R_y] &= 0
\end{align*}
\end{enumerate}
where $L_x(a) = xa$, $R_x(a) = ax$.
\end{block}
\begin{block}{Origin}
\begin{itemize}
\item Integrability of dynamical systems (Gelfand \& Dorfman).
\item Poisson brackets of hydrodynamic type (Balinsky \& Novikov).
\end{itemize}
\end{block}
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\begin{block}{Crucial fact}
Left-symmetricity $\Rightarrow$ Lie-admissibility:
$$
[x,y] = xy - yx
$$
satisfies the Jacobi identity.
\end{block}
\begin{block}{Classification of finite-dimensional simple Novikov algebras
over an algebraically closed field}
Zelmanov (1987) $p=0$: there are no non-trivial algebras.
Osborn (1992) $p>2$: for any such non-trivial algebra $A$, \\
\hskip 105pt $A^{(-)} \simeq W_1(n)$.
Xu (1996) $p>2$: described completely.
\end{block}
\uncover<2->{
\begin{block}{Reminder: Zassenhaus algebra}
$W_1(n) = \langle e_{-1}, e_0, e_1, \dots, e_{p^n-2} \rangle$
$[e_i,e_j] = \Big( \binom{i+j+1}j - \binom{i+j+1}i \Big) e_{i+j}$.
\end{block}
}
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\begin{block}{2-sided Alia algebras (Dzhumadil'daev, 2009)}
Alia = \textbf Anti \textbf{Li}e-\textbf admissible
\begin{align*}
[x,y]z + [z,x]y + [y,z]x &= 0 \\
z[x,y] + y[z,x] + x[y,z] &= 0
\end{align*}
$$
\begin{array}{l}
\text{commutative} \\
\text{Lie} \\
\text{Novikov} \\
\text{LR}
\end{array}
\Rightarrow \text{2-sided Alia} \Rightarrow \text{Lie-admissible}
$$
\end{block}
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\begin{block}{What are commutative $2$-cocycles?}
An algebra $A$ is 2-sided Alia iff $L = A^{(-)}$ is a Lie algebra and
multiplication in $A$ is given by
$$
xy = [x,y] + \varphi(x,y)
$$
where $\varphi: L \times L \to L$ is a \textbf{commutative $2$-cocycle} on $L$, i.e.:
\begin{enumerate}
\item $\varphi$ is symmetric
\item $\varphi([x,y],z) + \varphi([z,x],y) + \varphi([y,z],x) = 0$
\end{enumerate}
\medskip
\hskip -10pt $Z^2_{comm}(L)$ = the space of all $K$-valued commutative $2$-cocycles on $L$.
\end{block}
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\begin{block}{Theorem (Dzhumadil'daev \& Zusmanovich, 2010)}
A finite-dimensional simple Lie algebra over an algebraically closed field,
$p \ne 2,3$, possesses nonzero commutative $2$-cocycles iff it is isomorphic to
$sl(2)$ or $W_1(n)$.
\medskip
$\dim Z^2_{comm}(sl(2)) = 5$.
$Z^2_{comm}(W_1(n)) \simeq O_1(n)^*$.
\end{block}
}
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\begin{block}{A subtle point}
$p=3$.
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\begin{block}{Question}
$p=2$?
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}
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\begin{center}
{\Huge That's all. Thank you.}
\bigskip
Slides at \texttt{http://justpasha.org/math/coimbra.pdf}
\end{center}
}
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