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\title{A small step in classification of \\ simple Lie algebras in characteristic $2$}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{June 12, 2015}
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\titlepage
Partially based on \textsf{arXiv:1410.3645} (joint with Alexander Grishkov)
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\begin{frame}{Classification problem of simple Lie algebras}
\begin{itemize}
\item $p=0$: classic (Killing, E. Cartan, \dots)
\item $p>3$: a 3-volume set by H. Strade (the last one appeared in 2012)
\item $p=2,3$: open
\end{itemize}
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\begin{frame}{Toral rank}
\begin{block}{First surprise}
The $3$-dimensional simple Lie algebra $S$
$$
[e,h] = e, \quad [f,h] = f, \quad [e,f] = h
$$
has absolute toral rank $2$. Toral elements:
$h + e + e^{[2]}$, $h + f + f^{[2]}$.
\end{block}
\begin{block}{Theorem (Skryabin 1998)}
There are no finite-dimensional simple Lie algebras over an algebraically closed
field of absolute toral rank $1$.
\end{block}
\begin{block}{Ongoing project (Grishkov \& Premet)}
Absolute toral rank $2$?
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\begin{frame}{Toral rank (cont.)}
\begin{block}{Theorem (Grishkov \& Zusmanovich, 2014)}
A finite-dimensional simple Lie algebra over an algebraically closed field,
of absolute toral rank 2, and having a Cartan subalgebra of
toral rank 1, is isomorphic to $S$.
\end{block}
\begin{block}{Next goal}
Simple Lie algebras having a Cartan subalgebra of toral rank $1$.
\end{block}
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\begin{frame}{Ingredients of the proof}
\begin{itemize}
\item
Description of simple Lie algebras having a Cartan subalgebra of toral
rank $1$ as certain filtered deformations (Skryabin 1998).
\item
Computation of these filtered deformations in some cases.
\item
Low-degree cohomology of $S \otimes$ (commutative associative algebra).
\item
Dealing with a certain family of $15$-dimensional simple Lie algebras.
\item GAP.
\end{itemize}
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\begin{frame}{A family of $15$-dimensional algebras}
A $2$-parameter deformation of
$$
S \otimes \mathcal O_1(2) + f^{[2]} \otimes \langle 1,x \rangle + \partial
$$
(only those products are listed which differ in the deformed algebra).
\bigskip
Some properties:
\begin{itemize}
\item simple;
\item of absolute toral rank $3$;
\item $\Homol^2(L,K) = 0$;
\item do not possess symmetric invariant bilinear forms;
\item $p$-envelope coincides with derivation algebra and is of dimension $19$;
\item subalgebra generated by absolute zero divisors is of dimension $7$.
\end{itemize}
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\begin{frame}{A family of $15$-dimensional algebras}
{\small
\begin{alignat*}{5}
&&&\{e \otimes 1,\> &e &\otimes x &\} \>&=&\> f^{[2]} &\otimes \alpha x
\notag \\
&&&\{e \otimes 1,\> &e &\otimes x^{(2)} &\} \>&=&\> f^{[2]} &\otimes \beta 1
\notag \\
&\{e \otimes 1,\> &e \otimes x^{(3)} \} \>=\>\>
&\{e \otimes x,\> &e &\otimes x^{(2)} &\} \>&=&\> f^{[2]} &\otimes \beta x + \partial
\notag \\
&&&\{e \otimes x,\> &e &\otimes x^{(3)} &\} \>&=&\> h &\otimes 1
\notag \\
&&&\{e \otimes x^{(2)},\> &e &\otimes x^{(3)} &\} \>&=&\> h &\otimes x
\notag \\
&&&\{e \otimes x,\> &h &\otimes x^{(3)} &\} \>&=&\> f &\otimes 1
\notag \\
&&&\{e \otimes x^{(2)},\> &h &\otimes x^{(3)} &\} \>&=&\> f &\otimes x
\label{eq-15} \\
&&&\{e \otimes x^{(3)},\> &h &\otimes x^{(3)} &\} \>&=&\> f &\otimes x^{(2)}
\notag \\
&\{e \otimes x,\> &f \otimes x^{(3)} \} \>=\>\>
&\{e \otimes x^{(3)},\> &f &\otimes x &\} \>&=&\> f^{[2]} &\otimes 1
\notag \\
&\{e \otimes x^{(2)},\> &f \otimes x^{(3)} \} \>=\>\>
&\{e \otimes x^{(3)},\> &f &\otimes x^{(2)} &\} \>&=&\> f^{[2]} &\otimes x
\notag \\
&&&\{h \otimes x,\> &h &\otimes x^{(3)} &\} \>&=&\> f^{[2]} &\otimes 1
\notag \\
&&&\{h \otimes x^{(2)},\> &h &\otimes x^{(3)} &\} \>&=&\> f^{[2]} &\otimes x
\notag \\
&&&\{e \otimes 1,\> &\partial& &\} \>&=&\> f &\otimes \alpha x^{(3)}
\notag
\end{alignat*}
}
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\begin{frame}{``Commutative'' cohomology}
``Commutative'' Lie algebras:
$$
[x,y] = [y,x] \quad (\text{instead of }[x,x] = 0)
$$
+ Jacobi identity.
\bigskip
A natural cohomology in this class: in the Chevalley--Eilenberg complex,
replace alternating cochains by symmetric ones.
\bigskip
\begin{block}{Question}
Derived functor? Universal enveloping algebra?
\end{block}
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{\Huge That's all. Thank you.}
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Slides at \texttt{http://justpasha.org/math/porto.pdf}
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