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\title{On Lie algebras in characteristic $2$}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Institute of Mathematics, Prague
\\
March 2, 2022
}
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{\small
\begin{center}
Partially based on:
\end{center}
\vskip -10pt
A. Grishkov, H. Guzzo Jr., M. Rasskazova, P. Zusmanovich,
\emph{On simple $15$-dimensional Lie algebras in characteristic $2$},
J. Algebra \textbf{593} (2022), 295--318.
}
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\begin{frame}{Classification of simple Lie algebras}
$\bm{p=0}$:
\\
Killing, Cartan, Dynkin, ...
\bigskip
$\bm{p>3}$:
\\
Witt, Kostrikin, Shafarevich, Block, Wilson, Premet, Strade, ...
\bigskip
$\bm{p=2,3}$:
\\
?
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\begin{frame}{Tori, toral rank}
\textbf{Toral elements in the $p$-envelope}: $x^{[p]} = x$.
\medskip
\textbf{Torus}: A subalgebra consisting of toral elements.
\medskip
\textbf{(Absolute) toral rank}: Maximal dimension of a torus.
\bigskip
\begin{block}{Example ($p=2$)}
The simple $3$-dimensional Lie algebra $\ess$
$$
[e,h] = e , \quad [f,h] = f , \quad [e,f] = h
$$
has absolute toral rank $2$: $\langle h + e + e^{[2]}, h + f + f^{[2]} \rangle$
is a torus.
\end{block}
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\begin{frame}{The $15$-dimensional Skryabin algebra}
\textbf{From now on, we assume $p=2$.}
\vskip 25pt
A simple $15$-dimensional algebra defined by Skryabin (1998) appeared in the
classification of simple Lie algebras of absolute toral rank $2$ having a Cartan
subalgebra of toral rank $1$ (Grishkov--Zusmanovich, 2015).
\medskip
It is a deformation of a semisimple Lie algebra
$$
\ess \otimes \mathcal O_1(2) + f^{[2]} \otimes \langle 1,x \rangle + \partial ,
$$
where $\mathcal O_1(2)$ is the $4$-dimensional divided powers algebra over an
indeterminate $x$, $\partial$ its standard derivation.
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\begin{frame}{The $15$-dim. Skryabin algebra, multiplication table}
The $2$-parametic family $\mathscr L(\beta,\delta)$:
{\tiny
\begin{equation*}
\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
&b_2&b_3&b_4&b_5&b_6&b_7&b_8&b_9&c_1&c_2&c_3&c_4&c_5&d \\ \hline\hline
b_1&b_3&b_1&\beta c_5&b_6&b_4&\delta c_4&b_9&b_7&\delta c_5 + d&c_3&c_1&b_2&b_5&\beta c_2
\\ \hline
b_2&&b_2&b_6&0&b_5&b_9&0&b_8&c_3&0&c_2&0&0&0 \\ \hline
b_3&&&b_4&b_5&0&b_7&b_8&0&c_1&c_2&0&0&0&0 \\ \hline
b_4&&&&0&0&\delta c_5 + d&c_3&c_1&b_3&c_4&b_2&b_5&0&b_1 \\ \hline
b_5&&&&&0&c_3&0&c_2&c_4&0&0&0&0&b_2 \\ \hline
b_6&&&&&&c_1&c_2&0&0&0&c_4&0&0&b_3 \\ \hline
b_7&&&&&&&0&0&b_6&c_5&b_5&b_8&c_2&b_4 \\ \hline
b_8&&&&&&&&0&c_5&0&0&0&0&b_5 \\ \hline
b_9&&&&&&&&&0&0&c_5&0&0&b_6 \\ \hline
c_1&&&&&&&&&&0&b_8&c_2&0&b_7 \\ \hline
c_2&&&&&&&&&&&0&0&0&b_8 \\ \hline
c_3&&&&&&&&&&&&0&0&b_9 \\ \hline
c_4&&&&&&&&&&&&&0&0 \\ \hline
c_5&&&&&&&&&&&&&&c_4 \\ \hline
\end{array}
\end{equation*}
}
$\mathscr L(\beta,\delta)$ is a $2$-parametric linear deformation of
$\mathscr L(\beta,\delta)$ by two non-trivial $2$-cocycles,
but $\mathscr L(\beta,\delta) \simeq \mathscr L(0,0)$.
\bigskip
Such deformations are called \emph{semitrivial}
(Bouarroudj--Grozman--Lebedev--Leites--Shchepochkina, 2015, 2020).
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\begin{frame}{The $15$-dim. Skryabin algebra, some properties}
\begin{itemize}
\item $\Homol^2(\mathscr L,K) = 0$
\item $\dim \Homol^3(\mathscr L,K) = 15$
\item $\dim \Homol^2(\mathscr L,\mathscr L) = 13$
\item Derivation algebra = $2$-envelope is of dimension $19$.
\item No nontrivial symmetric bilinear forms.
\item The sandwich subalgebra is of dimension $3$.
\item The absolute toral rank is $4$.
\item
Lots of $7$-dimensional simple subalgebras, each isomorphic either to the
Zassenhaus algebra, or to a Hamiltonian algebra.
\item
$Aut(\mathscr L)$ is isomorphic to the semidirect product of $K^\times$ and a
$7$-dimensional unipotent algebraic group.
\item
Lots of gradings (over $\mathbb Z$, $\mathbb Z \oplus \mathbb Z / 2\mathbb Z$,
$(\mathbb Z / 2\mathbb Z)^4$, ...).
\end{itemize}
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\begin{frame}{Comparison with Eick's list}
Eick (2010) produced a computer-generated list of simple Lie algebras over
$\GF(2)$ of dimension $\le 20$. There are $8$ $15$-dimensional algebras in the
list, and the Skryabin algebra is not there!
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{\Huge That's all. Thank you.}
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