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\documentclass{beamer}
\usepackage{mathrsfs}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\HomCycl}{HomCycl}
\DeclareMathOperator{\HomLie}{HomLie}
\DeclareMathOperator{\HomtwoNilp}{Hom2Nilp}
\def\liebrack {\ensuremath{[\,\cdot\, , \cdot\,]}}
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\title{Hom-Lie and post-Lie structures on \\ current and Kac-Moody algebras}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Group32 \\ Prague \\ July 13, 2018
}
\begin{document}
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Based on:
\begin{itemize}
\item arXiv:1805.00187 (joint with Abdenacer Makhlouf)
\item arXiv:1805.04267 (joint with Dietrich Burde)
\end{itemize}
\vskip 30pt
These slides are at \texttt{http://www1.osu.cz/$\sim$zusmanovich/}
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\begin{frame}{What is a Hom-Lie algebra?}
A triple $(L, \liebrack, \alpha)$, where $\alpha: L \to L$ is linear, and
$\liebrack$ is anticommutative and satisfies the \emph{Hom-Jacobi} identity:
$$
[x,y] = -[y,x]
$$
$$
[[x,y],\alpha(z)] + [[z,x],\alpha(y)] + [[y,z],\alpha(x)] = 0
$$
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\begin{frame}{A little history}
\begin{itemize}
\item
Aizawa \& Sato (1991): ``Hom-Witt'' $(W_1, \liebrack, \alpha)$
\begin{align*}
&[e_i,e_j] = \{i-j\}_q e_{i+j}, \text{ where } \{n\}_q = \frac{q^n - q^{-n}}{q - q^{-1}}
\\
&\alpha(e_i) = \frac{q^i + q^{-i}}{2} e_i
\end{align*}
\item Hartwig, Larsson \& Silvestrov (2006): Hom-Lie algebras
\medskip
{\Huge ...}
\medskip
\item many people (2006-today):
Hom-associative:
$$
(xy)\alpha(z) = \alpha(x)(yz)
$$
and Hom-everything
\end{itemize}
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\begin{frame}{Hom-Lie structures}
A problem: given a Lie algebra $L$, determine all $\alpha$'s turning it into
a Hom-Lie algebra.
\medskip
Done for:
\begin{itemize}
\item
Finite-dimensional semisimples, $p=0$ (Jin \& Li 2008 and Xie, Jin \& Liu 2015)
\\
Answer: nontrivial structures only in the case of $sl(2)$.
\item
Witt algebra, infinite-dimensional algebras of Cartan type, loop algebras
(Xie \& Liu 2017).
\item
Current algebras and affine Kac-Moody algebras
(Makhlouf \& Zusmanovich 2018)
\\
Answer for Kac-Moody: ``almost trivial''.
\end{itemize}
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\begin{frame}{Hom-Lie structures in $p>0$}
Nothing is done in $p>0$.
\bigskip
\begin{block}{Conjecture}
If a simple finite-dimensional Lie algebra, $p>0$, admits a
nontrivial Hom-Lie structure, then it is isomorphic either to a 3-dimensional
simple algebra or to a (form of) Zassenhaus algebra.
\end{block}
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\begin{frame}{Hom-Lie structures on tensor products}
\begin{block}{Theorem}
Let $(\mathscr P, \mathscr P^!)$ be a (commutative, anticommutative) pair of
binary quadratic Koszul dual operads, $A$ an algebra over $\mathscr P$, $B$ an
algebra over $\mathscr P^!$. Then:
\begin{align*}
\HomLie\big((A &\otimes B)^{(-)}\big) \\
\>\simeq\>
&\HomLie(A) \otimes \HomCycl(B) + \HomtwoNilp(A) \otimes \End(B) \\
+\> &\HomCycl(A) \otimes \HomLie(B) + \End(A) \otimes \HomtwoNilp(B) .
\end{align*}
\end{block}
\vskip -10pt
\begin{alignat*}{2}
&\HomLie(A) &\>= &\text{ the space of all Hom-Lie structures on } A: \\
& & &(xy)\alpha(z) + (zx)\alpha(y) + (yz)\alpha(x) = 0
\\
&\HomCycl(A) &\>= &\text{ the space of all Hom-cyclic structures on } A: \\
& & &(xy)\alpha(z) = (zx)\alpha(y)
\\
&\HomtwoNilp(A) &\>= &\text{ the space of all Hom-2-nilpotent structures on } A:
\\
& & &(xy)\alpha(z) = 0
\end{alignat*}
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\begin{frame}{When Hom-Lie structures form a Jordan algebra?}
\begin{block}{Observation 1 (Xie, Jin \& Liu 2015, Xie \& Liu 2017)}
In all known cases so far, Hom-Lie structures on a Lie algebra form a Jordan
algebra with respect to
$$
\alpha * \beta = \frac 12 (\alpha \circ \beta + \beta \circ \alpha)
$$
\end{block}
\begin{block}{Observation 2 (Makhlouf \& Zusmanovich 2018)}
For some current Lie algebras $L \otimes A$ this is not so.
\end{block}
\begin{block}{Question}
When Hom-Lie structures form a Jordan algebra? Which Jordan algebras arise in
this way?
\end{block}
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\begin{frame}{Commutative post-Lie structures}
A problem: given a Lie algebra $L$, determine all bilinear products
$\cdot: L \times L \to L$ such that:
\begin{align*}
x \cdot y &= y \cdot x \\
[x,y] \cdot z &= x \cdot (y \cdot z) - y \cdot (x \cdot z) \\
x \cdot [y,z] &= [x \cdot y, z] + [y, x \cdot z]
\end{align*}
Done for:
\begin{itemize}
\item finite-dimensional perfect algebras, $p=0$ (Burde \& Moens 2016)
\\
Answer: trivial
\item free nilpotent algebras (Burde, Moens \& Dekimpe 2017)
\item Witt algebras and related (Tang 2017, Tang \& Yang 2018)
\\
Answer: trivial
\item loop algebras and affine Kac-Moody algebras (Burde \& Zusmanovich 2018)
\\
Answer: ``almost trivial''
\end{itemize}
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{\Huge
That's all. Thank you.
}
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