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\begin{document}
\title{
Questions about simple Lie algebras in characteristic $2$ by Alexander Grishkov
}
\author{Pasha Zusmanovich}
\address{}
\email{pasha.zusmanovich@gmail.com}
\date{first written February 3, 2021; last updated September 5, 2021}
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The ground field is assumed algebraically closed of characteristic $2$, unless
stated otherwise, and all algebras and modules assumed to be finite-dimensional.
$W_1$ and $H_2$ are $7$-dimensional simple Lie algebras, the Zassenhaus algebra
and the Hamiltonian algebra, respectively (see \cite{gg-2010} and \cite{gga}),
the Skryabin algebra is a certain simple $15$-dimensional algebra (see
\cite{15dim}).
\section{Lie algebras of low dimension}
\subsection{} (asked in \cite{gga}).
Describe irreducible representations of $W_1$ and $H_2$.
\subsection{Elements in the $15$-dimensional Skryabin algebra}
For the $2$-envelope of the $15$-di\-men\-si\-o\-nal Skryabin algebra,
describe conjugacy classes of the following elements:
\begin{enumerate}
\item toral elements;
\item elements satisfying $x^{[2]} \in \langle x \rangle$;
\item elements satisfying $x^{[2^n]} = 0$, for all $n$;
\end{enumerate}
\subsection{Gradings}
Describe gradings of the $7$-dimensional Hamiltonian algebra and the
$15$-di\-men\-si\-o\-nal Skryabin algebra (the latter question is asked in
\cite{15dim}). The $\mathbb Z / 2\mathbb Z$-gradings probably could be handled
with the approach from \cite{krutov-leb}.
\section{Classification of simple Lie algebras}
\subsection{}
Describe simple Lie algebras of dimension $7$. Conjecture: they are isomorphic
either to $W_1$, or to $H_2$. For algebras of absolute toral rank $3$, this is proved in \cite{gga}.
\subsection{}
For each known simple Lie algebra, compute its absolute toral rank and the
automorphism group.
\subsection{} (posed in \cite{gg}). Let $L$ be a Lie algebra over a field of
characteristic $p>0$ such that $\Der(L)$ contains a subalgebra $S$ isomorphic
to $\Sl_2$ (note that if $p=2$, then $\Sl_2$ is nilpotent). Let us call the
pair $(S,L)$ semisimple, if $L$ decomposed as an $S$-module as
$$
L = \bigoplus_i S_i \oplus \bigoplus_j V_j ,
$$
where each $S_i$ is a submodule of $M_2(K)$, and each $V_j$ is an irreducible
$S$-module.
Conjecture: $L$ is of classical type if and only if there exists a semisimple
pair $(L,S)$.
\subsection{Simple Lie algebras of toral rank $1$} Describe simple Lie algebras
$L$ having a Cartan subalgebra of toral rank $1$ in $L$. According to \cite[Theorem 6.3]{skryabin}, they are
either Zassenhaus, or Hamiltonian, or filtered deformations of semisimple Lie
algebras $G$ of the form
\begin{equation*}
S \otimes \mathcal O_1(n) \subset G \subseteq
\Der(S) \otimes \mathcal O_1(n) + K\partial ,
\end{equation*}
where either $n=2$ and $S \simeq W_1^\prime(m)$, or $n=1$ and
$S \simeq H_2^{\prime\prime}(m_1,m_2)$. In \cite{GZ} those deformations are
computed in the simplest case when $S = W_1^\prime(2)$, the $3$-dimensional
simple algebra. So the question is reduced to computation of those deformations in
general case.
Which of those simple algebras admit a thin decomposition?
\subsection{} Classify finite-dimensional simple Lie $2$-algebras.
\subsection{} (asked in \cite{15dim}). Classify finite-dimensional simple Lie
algebras having a $\mathbb Z$-grading with all homogeneous components of
dimension $<3$.
\section{Root space decompositions}
\subsection{}
Conjecture. Let $L$ be a simple Lie algebra, $T$ a torus of the maximal
dimension in the $2$-envelope of $L$, and $T \cap L = 0$. Then in the root space decomposition
$L = \oplus_\alpha L_\alpha$, $\dim L_\alpha$ is a constant (i.e., does not
depend on $\alpha$).
\subsection{Thin decompositions. Derivations} (asked in \cite{15dim}).
Conjecture. If $L$ is a simple Lie algebra admitting a thin decomposition, then
$\Der(L) \simeq T \oplus L$.
\subsection{Thin decompositions. Classification} (asked in \cite{15dim}).
To classify simple Lie algebras over an algebraically closed field admitting
a \emph{thin decomposition}, i.e., when $\dim L = 2^n-1$, $\dim T = n$, the
roots are exactly $GF(2)^n \backslash (0,\dots,0)$, and $\dim L_\alpha = 1$ for any root
$\alpha$.
\subsection{Thin decompositions. Subalgebras}
Prove that any simple Lie algebra of dimension $>3$ over a field of
characteristic $2$ admitting a thin decomposition, has:
\\
a) a proper simple graded subalgebra (with respect to this decomposition);
\\
b) a graded subalgebra isomorphic either to $W_1$ or to $H_2$.
\subsection{Thin decompositions. Modules}
Conjecture. Let $L$ be a simple Lie algebra of dimension $2^n-1$ with a thin
decomposition, $V$ an irreducible $2^n$-dimensional $L$-module. Assume that
there is a simple Lie algebra with thin decomposition of the form $L \oplus V$,
where $L$ is a subalgebra, the multiplication between $L$ and $V$ is given by
the action of $L$ on $V$, and the multiplication between elements of $V$ is
given by the map $f: V \times V \to L$. The the map $f$ is determined uniquely.
Study this situation for $L=W_1$ or $H_2$ (the Skryabin algebra arises in this
way from $H_2$).
\subsection{Variety of algebras with a thin decomposition}
Let $L$ be a Lie algebra with thin decomposotion, with multiplication defined
by $[e_g,e_h] = f(g,h)e_{g+h}$. Study the variety of all possible functions $f$
defining a thin Lie algebra, from the algebro-geometric viewpoint. What are
irreducible components? Which simple Lie algebras are ``generic'' in this sense?
\begin{thebibliography}{GGRZ}
\bibitem[GG1]{gg} A. Grishkov, M. Guerreiro,
\emph{Simple classical Lie algebras in characteristic $2$ and their gradations.
I},
Intern. J. Math. Game Theory Algebra \textbf{13} (2003), no.3, 239--252.
\bibitem[GG2]{gg-2010} \bysame{}, \bysame,
\emph{On simple Lie algebras of dimension seven over fields of characteristic $2$}, S\~{a}o Paulo J. Math. Sci. \textbf{4} (2010), no.1, 93--107.
\bibitem[GGA]{gga} \bysame, \bysame, W.F. de Araujo,
\emph{On the classification of simple Lie algebras of dimension seven over
fields of characteristic $2$},
S\~ao Paulo J. Math. Sci. \textbf{14} (2020), no.2, 703--713.
\bibitem[GGRZ]{15dim} \bysame, H. Guzzo Jr., M. Rasskazova, P. Zusmanovich,
\emph{On simple $15$-dimensional Lie algebras in characteristic $2$},
manuscript.
\bibitem[GZ]{GZ} \bysame{}, P. Zusmanovich,
\emph{Deformations of current Lie algebras. I. Small algebras in characteristic
$2$},
J. Algebra \textbf{473} (2017), 513--544.
\bibitem[KL]{krutov-leb} A. Krutov, A. Lebedev,
\emph{On gradings modulo 2 of simple Lie algebras in characteristic $2$},
SIGMA \textbf{14} (2018), 130.
\bibitem[S]{skryabin} S. Skryabin,
\emph{Toral rank one simple Lie algebras of low characteristics},
J. Algebra \textbf{200} (1998), no.2, 650--700.
\end{thebibliography}
\end{document}
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