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% created:      pasha jun 11-13 2016
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\title{Cohomology of tensor products}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{Inregrable Systems and Quantum Symmetries, Prague \\ June 14, 2016}


\begin{document}

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\titlepage
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\begin{frame}{A Lie-algebraic zoo. I}

\textbf{Current Lie algebras}

\bigskip

$C^\infty (M, \mathfrak g) \simeq \mathfrak g \otimes C^\infty(M, \mathbb R)$

\bigskip

\begin{block}{Generalization}
$L \otimes A$
\\
$L$ is a Lie algebra, $A$ an associative commutative algebra
\\
$[x \otimes a, y \otimes b] = [x,y] \otimes ab$
\end{block}

\bigskip

\begin{block}{Closely related}
Kac--Moody algebras
\end{block}

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\begin{frame}{A Lie-algebraic zoo. II}

\textbf{``Poisson brackets of hydrodynamic type'' 
\\
(S.P. Novikov et al.)}

\vskip 2cm

$[u_i(x),u_j(y)] = g_{ij}(u(x))\delta^\prime(x-y) +
\sum_{k=1}^n \frac{\partial u_k}{\partial x} b_k^{ij}(u(x))\delta(x-y)$

\end{frame}
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\begin{frame}{A Lie-algebraic zoo. III}

\textbf{Heisenberg--Virasoro, Schr\"odinger--Virasoro, etc.}

\bigskip

Certain extensions/generalizations of the Witt algebra

\medskip

\begin{block}{``Affinization'' of a Novikov algebra (Pei \& Bai, 2010--2011)}
$N \otimes \mathbb C[t,t^{-1}]$
\\
$[x \otimes t^n, y \otimes t^m] = ((n+1) xy - (m+1) yx) \otimes t^{n+m}$
\end{block}

\begin{block}{Generalization}
$A \otimes B$
\\
$A$ is a left Novikov algebra, $B$ a right Novikov algebra
\\
$[a \otimes b, a^\prime \otimes b^\prime] = 
aa^\prime \otimes bb^\prime - a^\prime a \otimes b^\prime b$
\end{block}

\begin{block}{Even more generalization}
$A, B$ are algebras over Koszul dual operads
\end{block}

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\begin{frame}{A Lie-algebraic zoo. IV}

\textbf{``Lie algebras generated by dynamical systems'' 
\\
(Vershik et al.)}

\bigskip

$C(X,\mathbb C) \otimes \mathbb C[t,t^{-1}]$
\\
$(X,T)$ is a dynamical system
\\
$[f \otimes t^n, g \otimes t^m] = 
(f \cdot (g \circ T^n) - g \cdot (f \circ T^m)) \otimes t^{n+m}$

\medskip

\begin{block}{Generalization}
$A \otimes B$
\\
$A, B$ are associative commutative algebras
\\
generators of $B$ act on $A$ by automorphisms
\\
$[a \otimes b, a^\prime \otimes b^\prime] = 
(a(a^\prime)^b - a^\prime(a)^{b^\prime}) \otimes bb^\prime$
\end{block}

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\begin{frame}{A Lie-algebraic zoo. V}

\textbf{(Some) Lie algebras of symmetries of differential equations}

\bigskip

\begin{block}{Symmetries of Khokhlov--Zabolotskaya, Boyer--Finley, etc., equations (O. Morozov, 2015)}

$C^{\infty}(\mathbb R, \mathbb R) \otimes \mathbb R[t]/(t^n) + 
\text{ tail of derivations}$

$
[f \otimes t^i , g \otimes t^j] = 
\Big(\sum_{k=0}^{i+j} \lambda_{ijk} f^{(k)} g^{(i+j-k)}\Big) \otimes t^{i+j}
$

\end{block}

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\begin{frame}{A Lie-algebraic zoo}

... and (much) more

\vskip 1.5cm

\begin{block}{Question}
What these algebras have in common?
\end{block}

\begin{block}{Answer}
$A \otimes B$ with a ``twisted'' multiplication.
\end{block}

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\begin{frame}{Now we have a zoo. What we can do with it?}

\begin{block}{Question}
To compute (co)homology and other invariants of such algebras.
\end{block}

\begin{block}{An attempt of answer}
Decompose the Chevalley--Eilenberg complex according to the Cauchy formula:
$$
\bigwedge\nolimits^n (A \otimes B) \simeq
\bigoplus_\lambda \Y_\lambda(A) \otimes \Y_{\lambda^\sim}(B)
$$
$\Y_\lambda$ = Young symmetrizer corresponding to the Young diagram $\lambda$
\end{block}

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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 
$Hom(Y_\lambda(L),?) \otimes Hom(Y_\lambda(A),?)$
}
\end{center}

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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 
$Hom(Y_\lambda(L),?) \otimes Hom(Y_\lambda(A),?)$
}
\end{center}

\end{frame}
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\begin{frame}{A plethora of questions. I}

\begin{block}{Question}
Why this miracle happens only for current Lie algebras?
\end{block}

\begin{block}{Question}
When some partial miracle may happen?
\end{block}

\begin{block}{A (very) partial answer}
\begin{itemize}

\item 
$\Homol^2$ for ``$A \otimes B$ over Koszul dual operads'' when 
``noncommutative $2$-cocycles'' on $A$ and $B$ are ``small''
(captures the Heisenberg-- and Schr\"odinger--Virasoro cases).

\item $\Homol^2$ for ``Lie algebras generated by dynamical systems''.

\item $\Homol^2$ for symmetry Lie algebras of some differential equations.

\end{itemize}
\end{block}

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\begin{frame}{A plethora of questions. II}

Algebras currently not in ``the zoo'' which I would like 
very much to include:

\begin{itemize}

\item Lax operator algebras (Krichever, Schlichenmaier, Sheinman)

\item The Gaudin algebras
$[S_i(x),S_j(y)] = \sum_k c_{ij}^k \frac{S_k(x) - S_k(y)}{x-y}$

\item The Onsager algebra

\item ``Lie algebras over noncommutative rings'' (Berenstein and Retakh)

\end{itemize}

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{\Huge 
That's all. Thank you.
}
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