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\title{Lie algebras with few subalgebras, cohomology and dual operads}
\author{Pasha Zusmanovich}
\institute{Tallinn University of Technology}
\date{February 22, 2013}
\begin{document}
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\titlepage
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{
\vskip2pt\hskip1pt\insertframenumber /\inserttotalframenumber
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\begin{frame}{Lie algebras with given properties of subalgebras}
\begin{block}{Question}
Study such Lie algebras.
\end{block}
\bigskip
\begin{block}{More precise questions}
\begin{itemize}
\item Lie algebras with ``few'' subalgebras.
\item Lie algebras with given properties of the lattice of subalgebras.
\item Minimal non-$\mathscr P$ Lie algebras, for some ``nice'' $\mathscr P$.
\end{itemize}
\end{block}
\end{frame}
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\begin{frame}{Lie algebras all whose proper subalgebras are $1$-dimensional}
\begin{block}{Exercise}
Over an algebraically closed field, every such finite-dimensional Lie algebra
is $2$-dimensional.
\end{block}
\uncover<2->{
\begin{block}{Theorem}
Over a perfect field of characteristic $0$ or $p>3$, every such
finite-dimensional Lie algebra is either $2$-dimensional, or is a form of
$\mathsf{sl}(2)$.
\end{block}
\textbf{Proof}:
Either follows from A. Premet (1987), or by classification theory.
}
\uncover<3->{
\begin{block}{An open question}
What about nonperfect fields and/or $p=2,3$?
\end{block}
}
\uncover<4->{
\begin{block}{A very hard open question}
What about such infinite-dimensional Lie algebras
(analogs of Tarski's monsters)?
\end{block}
}
\end{frame}
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\begin{frame}{Lie-algebraic analogs of Tarski's monsters?}
Possible approaches:
\begin{itemize}
\item First- (or higher?) order theory.
\item Girth.
\item (Absence of) identities.
\end{itemize}
\end{frame}
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\begin{frame}{Lie algebras all whose proper subalgebras are abelian}
\begin{block}{Exercise}
(i)
Over an algebraically closed field, every such nonabelian finite-dimensional
Lie algebra is either $2$-dimensional nonabelian, or $3$-dimensional nilpotent
(Heisenberg).
(ii)
Describe the structure of such nonsimple Lie algebras over any field.
\end{block}
\uncover<2->{
\begin{block}{Theorem}
Over a perfect field of characteristic $0$ or $p>3$, there are no such
nonabelian simple finite-dimensional Lie algebra of types
$B$-$D$, $G_2$ and $F_4$.
\textbf{Proof}:
By inspection of (associative) division algebras with involution.
\end{block}
\begin{block}{An open question}
What about other types?
\end{block}
}
\uncover<3->{
\begin{block}{A curious connection}
A.M. Vinogradov (2012): ``Assembling Lie algebras from Lieons''.
\end{block}
}
\end{frame}
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\begin{frame}{Lie algebras all whose proper subalgebras are solvable}
\begin{block}{Exercise}
Over an algebraically closed field of characteristic $0$, every such nonsolvable
finite-dimensional Lie algebra is isomorphic to $\mathsf{sl}(2)$.
\end{block}
\uncover<2->{
\begin{block}{Theorem}
Over an algebraically closed field of characteristic $p>3$, every such
nonsolvable finite-dimensional Lie algebra is an abelian extension of
$\mathsf{sl}(2)$ by modules of some specific type (decomposable into the direct
sum of no more than $2$ components, with each indecomposable component
of length $\le 3$, etc.).
\smallskip
\textbf{Proof}
(modulo
N. Jacobson (1958), A. Rudakov \& I. Shafarevich (1967), J. Schue (1969)):
Cohomological juggling (computing $Ext^1$, $H^2$, etc.) with
$\mathsf{sl}(2)$-modules.
\end{block}
}
\end{frame}
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\begin{frame}{Lie algebras with a maximal solvable subalgebra}
\begin{block}{Exercise}
Over an algebraically closed field of characteristic $0$, every such semisimple
finite-dimensional Lie algebra is isomorphic to $\mathsf{sl}(2)$.
\end{block}
\uncover<2->{
\begin{block}{Theorem}
Over an algebraically closed field of characteristic $p>5$, every such
semisimple finite-dimensional Lie algebra is isomorphic to an algebra
of the form
$$
S \otimes K[x_1, \dots, x_n]/(x_1^p, \dots, x_n^p) +
\text{``some tail of derivations''} ,
$$
where $S$ is isomorphic to $\mathsf{sl}(2)$, or the Zassenhaus algebra $W_1(n)$.
\smallskip
\textbf{Proof} (modulo B. Weisfeiler (1984)):
Computing deformations of algebras of this kind.
\end{block}
}
\end{frame}
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\begin{frame}{Current Lie algebras}
$L$ is a Lie algebra
$A$ is an associative commutative algebra
\bigskip
A \textbf{current Lie algebra} is a vector space
$L \otimes A$ under the bracket
$$
[x \otimes a, y \otimes b] = [x,y] \otimes ab
$$
where $x,y\in L$, $a,b\in A$.
\uncover<2->{
\begin{block}{Kac-Moody algebras}
$$
\mathfrak g \otimes \mathbb C[t,t^{-1}] + \mathbb C t\frac{d}{dt} + \mathbb C z
$$
$$
[x \otimes f, y \otimes g] = [x,y] \otimes fg + (x,y)Res \frac{df}{dt}g \> z
$$
where $\mathfrak g$ is a simple finite-dimensional Lie algebra,
$x,y\in \mathfrak g$, $f,g\in \mathbb C[t,t^{-1}]$,
$\form$ is the Killing form on $\mathfrak g$.
\end{block}
}
\end{frame}
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\begin{frame}{(Co)homology of current Lie algebras}
\begin{block}{Question}
What can be said about it?
\end{block}
\begin{block}{Answer}
A lot:
\medskip
B. Feigin (1970--1990s),
H. Garland \& J. Lepowsky (1976),
B. Feigin \& B. Tsygan (1983--1984),
J.-L. Loday \& D. Quillen (1984),
P. Hanlon (1986), ...
\end{block}
but ...
\end{frame}
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\begin{frame}{How to ``compute'' (co)homology of current Lie algebras ``in general''?}
Cauchy formula:
$$
\bigwedge\nolimits^n(L\otimes A) \simeq \bigoplus_{\lambda \vdash n} Y_\lambda(L) \otimes Y_{\lambda^\sim}(A)
$$
$Y_\lambda$ is a \textit{Young symmetrizer} associated with the Young diagram $\lambda$. \\
Examples:
\Yboxdim5pt
$Y_{\yng(1,1,1)} \hskip 9pt = \frac{1}{3!} \sum_{\sigma\in S_3} (-1)^\sigma \sigma$ \\
$Y_{\yng(3)} = \frac{1}{3!} \sum_{\sigma\in S_3} \sigma$ \\
$Y_{\yng(2,1)} \hskip 5pt = \frac{1}{3} (e + (12) - (13) - (123))$
\bigskip
$\lambda^\sim$ is obtained from $\lambda$ by interchanging rows and columns
\end{frame}
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\begin{frame}{How Young symmetrizers interact with the differential?}
% all arrows
\begin{diagram}[width=2.3em,height=2.5em]
&&&&&& \yng(1) &&&&&&& \\ %1
&&&&& \ldTo && \rdTo &&&&& \\ %2
&&&& \yng(1,1) &&&& \yng(2) &&&& \\ %3
&&& \ldTo && \rdTo\rdTo(5,2) & & \ldTo(6,2)\ldTo && \rdTo &&& \\ %4
&& \yng(1,1,1) &&&& \yng(2,1) &&&& \yng(3) && \\ %5
& \ldTo && \rdTo\rdTo(4,2)\rdTo(6,2)\rdTo(10,2) && \ldTo(6,2)\ldTo & \dTo & \rdTo\rdTo(6,2) && \ldTo(10,2)\ldTo(6,2)\ldTo(4,2)\ldTo && \rdTo & \\ %6
\yng(1,1,1,1) &&&& \yng(2,1,1) && \yng(2,2) && \yng(3,1) &&&& \yng(4) \\ %7
\dots &&&& \dots && \dots && \dots &&&& \dots
\end{diagram}
\begin{center}
{\tiny
each Young diagram $\lambda$ represents
$Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
}
\end{center}
\end{frame}
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\begin{frame}{How Young symmetrizers interact with the differential?}
% non-vanishing arrows (taken from low.tex)
\begin{diagram}[width=2.3em,height=2.5em]
&&&&&& \yng(1) &&&&&&& \\ %1
&&&&& \ldTo && \rdTo &&&&& \\ %2
&&&& \yng(1,1) &&&& \yng(2) &&&& \\ %3
&&& \ldTo && \rdTo & & \ldTo(6,2)\ldTo && \rdTo &&& \\ %4
&& \yng(1,1,1) &&&& \yng(2,1) &&&& \yng(3) && \\ %5
& \ldTo && \rdTo\rdTo(4,2) && \ldTo(6,2)\ldTo & \dTo & \rdTo && \ldTo(10,2)\ldTo(6,2)\ldTo(4,2)\ldTo && \rdTo & \\ %6
\yng(1,1,1,1) &&&& \yng(2,1,1) && \yng(2,2) && \yng(3,1) &&&& \yng(4) \\ %7
\dots &&&& \dots && \dots && \dots &&&& \dots
\end{diagram}
\begin{center}
{\tiny
each Young diagram $\lambda$ represents
$Hom(Y_\lambda(L),M) \otimes Hom(Y_\lambda(A),V)$
}
\end{center}
\end{frame}
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\begin{frame}{Lie algebras coming from Koszul dual operads}
\begin{block}{Fact}
If:
$A$ is an algebra over a binary quadratic operad $\mathscr P$,
$B$ is an algebra over the Koszul dual operad $\mathscr P^!$,
then:
$A \otimes B$ carries a Lie algebra structure under the bracket
$$
[a \otimes b, a^\prime \otimes b^\prime] =
aa^\prime \otimes bb^\prime - a^\prime a \otimes b^\prime b ,
$$
where $a,a^\prime \in A$, $b,b^\prime \in B$.
\end{block}
\begin{block}{Examples}
\begin{tabular}{|p{1.9cm}|p{3.9cm}|p{3.1cm}|}
\hline
\centering \textbf{operad} & \centering \textbf{dual operad} & \centering \textbf{Lie algebras}
\tabularnewline \hline
Lie & associative commutative & current Lie algebras \\ \hline
associative & associative & $\mathsf{gl}_n(A)$ \\ \hline
left Novikov & right Novikov & ... {\tiny stay tuned} ... \\ \hline
\end{tabular}
\end{block}
\end{frame}
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\begin{frame}{Novikov algebras and their affinizations}
\textbf{left Novikov algebra}:
\hskip7pt $[L_x,L_y] \hskip 2pt = L_{[x,y]}$; \hskip 1pt $[R_x,R_y] = 0$
\textbf{right Novikov algebra}:
$[R_x,R_y] = R_{[x,y]}$; $[L_x,L_y] \hskip 2pt = 0$
where $L_x(a) = xa$, $R_x(a) = ax$
\begin{block}{Affinization of a left Novikov algebra}
$$
N \otimes \mathbb C[t,t^{-1}]
$$
$$
[x \otimes t^m, y \otimes t^n] = \Big((m+1) xy - (n+1) yx\Big) \otimes t^{m+n}
$$
where $N$ is a left Novikov algebra, $x,y\in N$, $m,n\in \mathbb Z$.
\end{block}
\begin{block}{Particular cases}
\begin{itemize}
\item
``Poisson brackets of hydrodynamic type''
(I. Gelfand \& I. Dorfman (1979--1981), A. Balinskii \& S.P. Novikov (1985)).
\item
Schr\"odinger--Virasoro, Heisenberg--Virasoro (Y. Pei \& C. Bai (2010--2012)).
\item
Finite-dimensional simple Lie algebras over $p=2,3$ {\Large \textbf{?}}
\end{itemize}
\end{block}
\end{frame}
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\begin{frame}{(Co)homology of Lie algebras coming from dual operads}
\begin{block}{Question}
Express (co)homology and other invariants
(symmetric invariant bilinear forms, etc.) of $A \otimes B$
in terms of invariants of $A$ and $B$.
\end{block}
\begin{block}{Potential applications}
\begin{itemize}
\item
``Physics''
(central extensions, $2$-Lie algebras from (higher) gauge theory, ...)
\item
Structure theory of finite-dimensional Lie algebras in small characteristics.
\item
New invariants (cyclic cohomology-like?) of nonassociative algebras.
\end{itemize}
\end{block}
\end{frame}
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\begin{center}
{\Huge That's all. Thank you.}
\end{center}
\vskip30pt
\begin{center}
Slides at \texttt{http://justpasha.org/math/ihes.pdf}
\end{center}
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