%
% file: talk-covilha-homjord.tex
% purpose: talk at Covilha, Fri Nov 19 2021
% created: pasha nov 16-19 2021
%
\documentclass{beamer}
\newcommand{\vertbar}{\>|\>}
\DeclareMathOperator{\HomLie}{HomLie}
\setcounter{framenumber}{-1}
\setbeamertemplate{navigation symbols}{}
\title{When Hom-Lie structures form a Jordan algebra}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Algebra Seminar
\\
Universidade da Beira Interior, Covilh\~{a}
\\
November 19, 2021
}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\begin{center}
\small
Based on the manuscript
\\
\texttt{https://web.osu.cz/$\sim$Zusmanovich/papers/homlie-jordan.pdf}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setbeamertemplate{headline}
{
\vskip2pt\hskip1pt\insertframenumber /\inserttotalframenumber
}
\setbeamertemplate{frametitle}
{
\vskip8pt\insertframetitle
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Hom-Lie structures}
A Hom-Lie structure on a Lie algebra $L$ is a linear map
$\varphi: L \to L$ such that
$$
[[x,y],\varphi(z)] + [[z,x],\varphi(y)] + [[y,z],\varphi(x)] = 0 .
$$
$\HomLie(L)$ -- the vector space of all Hom-Lie structures on $L$.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Hom-Lie structures on ``interesting'' Lie algebras}
\begin{itemize}
\item
Classical simple: trivial except of $sl(2)$
(Jin--Li 2008, Xie--Jin--Liu 2015)
\item
Kac--Moody: ``almost'' trivial (Makhlouf--Zusmanovich 2018)
\item
Lie algebras of vector fields: trivial except of the Witt algebra
(Xie--Liu 2017)
\item
Generalized Witt algebra $W_G$, $G \subseteq F^\times$,
$$
[e_\alpha, e_\beta] = (\beta - \alpha) e_{\alpha+\beta}
$$
$\HomLie(W_G) \simeq K[G]$, spanned by $e_\alpha \mapsto e_{\alpha + \sigma}$.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Hom-Lie structures as a Jordan algebra}
\begin{block}{Observation}
In all these cases, $\HomLie(L)$ is closed with respect to the
``Jordan product''
$$
\frac 12 (\varphi \circ \psi + \psi \circ \varphi)
$$
\end{block}
\begin{block}{Question}
For which Lie algebras $L$ this is true?
\end{block}
\begin{block}{Answer}
Not for all $L$.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{An example}
$$
L = \langle x,y \vertbar [x,y] = x \rangle \otimes K[t]
$$
$$
\varphi \otimes 1, \psi \otimes \alpha \in \HomLie(L)
$$
$$
(\varphi \otimes 1) \circ (\psi \otimes \alpha)
+ (\psi \otimes \alpha) \circ (\varphi \otimes 1) \notin \HomLie(L)
$$
$$
\varphi: \>
\begin{aligned}
x &\mapsto y \\
y &\mapsto 0
\end{aligned}
\hskip 20pt
\psi: \>
\begin{aligned}
x &\mapsto x \\
y &\mapsto 0
\end{aligned}
$$
$$
\alpha(t^i) = t^{2i}
$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A criterion}
The following are equivalent:
\begin{itemize}
\item $\HomLie(L)$ is a Jordan algebra;
\item $\HomLie(L)$ is closed with respect to squares;
\item $\HomLie(L)$ is closed with respect to polynomials;
\item For any $\varphi, \psi \in \HomLie(L)$,
$$
[F(x,y),z] + [F(z,x),y] + [F(y,z),x] = 0
$$
where
$$
F(x,y) = [\varphi(x),\psi(y)] + [\psi(x),\phi(y)] .
$$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Connection with CYBE}
\begin{block}{Known fact}
$\varphi: L \to L$ is an $R$-matrix iff
\begin{equation}\tag{$\star$}
[F(x,y),z] + [F(z,x),y] + [F(y,z),x] = 0
\end{equation}
where
$$
F(x,y) = [\varphi(x),\varphi(y)] - \varphi([\varphi(x),y] + [x,\varphi(y)]) .
$$
\end{block}
\bigskip
%\begin{block}{Remark}
F(x,y) = 0 is the (famous) QYBE.
%\end{block}
\bigskip
\begin{block}{Question}
Study skew-symmetric solutions of ($\star$) on various Lie algebras.
\end{block}
\bigskip
Symmetric solutions of ($\star$) appear in the study of Lie-admissible
power-associative algebras (Benkart 1984).
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{What happens when $HomLie(L)$ is a Jordan algebra}
\begin{block}{Theorem}
Let $L$ be a finite-dimensional Lie algebra over an algebraically closed field
such that $\HomLie(L)$ forms a Jordan algebra. Then one of the following holds:
\begin{enumerate}
\item[(*)] $HomLie(K) \simeq K$.
\item[($\diamondsuit$)]
$L = A \oplus B$ (as vector space), $A, B \ne 0$, $[[A,A],B] = 0$,
$[[B,B],A] = 0$.
\item[($\heartsuit$)]
$0 \ne A \subseteq B \subset L$ (vector spaces), $\dim A + \dim B = \dim L$,
$[[A,A],B] = 0$, $[[B,B],A] = 0$.
\end{enumerate}
\end{block}
\bigskip
$W_G$ in characteristic zero provides an infinite-dimensional counterexample to
this theorem: neither of the conditions is satisfied.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{$\diamondsuit$ and $\heartsuit$}
\begin{block}{Question}
Study the properties $\diamondsuit$ and $\heartsuit$.
\end{block}
\begin{block}{Conjecture 1}
$\heartsuit$ has something to do with subalgebras of codimension $1$.
\end{block}
\begin{block}{Conjecture 2}
\begin{enumerate}[\upshape(i)]
\item
If $L$ satisfies $\diamondsuit$, the $L/Rad(L)$ is the direct sum of $sl(2)$'s.
\item
If $L$ satisfies $\heartsuit$, then $L/Rad(L)$ is the direct sum of $sl(2)$'s
and the Zassenhaus algebras.
\end{enumerate}
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% don't include this final frame into the total count;
% TeX twice for that!
\newcounter{finalframe}
\setcounter{finalframe}{\value{framenumber}}
\setbeamertemplate{headline}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{center}
{\Huge That's all. Thank you.}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{framenumber}{\value{finalframe}}
\end{document}
% eof