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\title{On simple $15$-dimensional Lie algebras in characteristic $2$}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Centre for Mathematics
\\
University of Coimbra
\\
Algebra and Combinatorics Seminar
\\
November 17, 2021
}
\begin{document}
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\begin{frame}
\titlepage
\begin{center}
\small
Joint work with \\ Alexander Grishkov, Henrique Guzzo Jr., and Marina Rasskazova
\\
\texttt{https://web.osu.cz/$\sim$Zusmanovich/papers/15dim.pdf}
\end{center}
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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
$\bm{p=2,3}$:
\\
?
\end{frame}
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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}
\end{frame}
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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 $1$ (2015, Grishkov \& Zusmanovich).
\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}
{\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*}
}
\end{frame}
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\begin{frame}{The $15$-dim. Skryabin algebra, some properties}
\begin{itemize}
\item $\Homol^2(L,K) = 0$
\item $\dim \Homol^3(L,K) = 15$
\item $\dim \Homol^2(L,L) = 13$
\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
$Aut(L)$ 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}
\end{frame}
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\begin{frame}{Thin decomposition}
Let $\dim L = 2^n-1$, the absolute toral rank is $n$, $T$ is a torus of
dimension $n$ in the $2$-envelope such that $L \cap T = 0$. Let the root
space decomposition of $L$ with respect to $T$ is of the form
\begin{equation*}
L = \bigoplus_{\substack{\alpha \in \GF(2)^n \\ \alpha \ne (0,\dots,0)}}
\langle e_\alpha \rangle .
\end{equation*}
Such decomposition is called \emph{thin}.
The multiplication table in the basis $\{e_\alpha\}$ is of the form
$$
[e_\alpha, e_\beta] = \begin{cases}
e_{\alpha + \beta} \\
0
\end{cases}
$$
\begin{block}{Fact}
Over $\GF(2)$, the Skryabin algebra admits a thin decomposition with respect to
each of the $26,880$ $4$-dimensional tori in the $2$-envelope.
\end{block}
\end{frame}
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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!
\end{frame}
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\begin{frame}{Some further questions}
\begin{itemize}
\item Classify simple Lie algebras admitting a thin decomposition.
\item Prove that in the ``most'' of the cases a simple Lie algebra contains
a simple $7$-dimensional subalgebra.
\end{itemize}
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\begin{center}
{\Huge That's all. Thank you.}
\end{center}
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