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\title{Commutative cohomology in characteristic $2$}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
Institute of Mathematics, Prague \\
November 6, 2019
}
\begin{document}
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\titlepage
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\begin{frame}{Commutative Lie algebras}
\begin{center}
\textbf{Standing assumption: $p=2$}
\end{center}
$$
[x,y] = [y,x]
$$
$$
[[x,y],z] + [[z,x],y] + [[y,z],x] = 0
$$
\bigskip
\begin{center}
Lie \quad$\subset$\quad commutative Lie \quad$\subset$\quad Leibniz
\\
\hspace{-186pt} $[x,x] = 0$
\end{center}
\end{frame}
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\begin{frame}{Commutative cohomology}
$
0 \to \sym^0(L,M) \overset{\dcobound}{\to} \sym^1(L,M) \overset{\dcobound}{\to}
\sym^2(L,M) \overset{\dcobound}{\to} \dots
$
\vspace{23pt}
$\sym^n(L,M)$ = the space of $n$-linear \emph{symmetric} maps
$L \times \dots \times L \to M$
\begin{align*}
\dcobound \varphi(x_1, \dots, x_{n+1}) & \\
&= \sum_{1 \le i < j \le n+1}
\varphi([x_i,x_j],x_1,\dots,\widehat{x_i},\dots,\widehat{x_j},\dots,x_{n+1})
\\
&+ \hspace{16pt} \sum_{i=1}^{n+1} \hspace{16pt} x_i \bullet \varphi(x_1,\dots,\widehat{x_i},\dots,x_{n+1}) .
\end{align*}
\end{frame}
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\begin{frame}{Why bother?}
\begin{enumerate}[1)]
\item
Operadic viewpoint: algebras should be defined by multilinear identities.
\item
New phenomena in cohomology, similar to Lie superalgebras.
\item
New invariant of (ordinary) Lie algebras.
\item
Appears naturally in the context of classification of simple Lie algebras.
\end{enumerate}
\end{frame}
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\begin{frame}{Some spectral sequences}
(after Friedrich Wagemann)
\bigskip
\begin{itemize}
\item
Hochschild--Serre spectral sequence
\medskip
$0 \to I \to L \to L/I \to 0$
\medskip
$E^{pq}_2 = \Homol^p_{comm}(L/I, \Homol^q_{comm}(I,M)) \>\Rightarrow\>
\Homol^{p+q}_{comm}(L,M)$
\bigskip
\item
Spectral sequences relating commutative cohomology with
Chevalley--Eilenberg cohomology and with Leibniz cohomology
\end{itemize}
\end{frame}
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\begin{frame}{Cup product}
$
(\varphi \smile \psi)(x_1,\ldots, x_{p+q}) =
\sum_{\text{shuffles}} \varphi(x_{i_1}, \ldots, x_{i_p})\cdot \psi(x_{j_1},\ldots,x_{j_q})
$
\bigskip
$\smile$ turns $\Homol^*_{comm}(L,K)$ into an associative graded ring.
\end{frame}
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\begin{frame}{Some computations}
\begin{center}
\includegraphics[scale=0.08]{lopatkin-heisenberg.jpg}
\end{center}
\end{frame}
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\begin{frame}{$2$-dimensional nonabelian}
$L = \langle a,b \vertbar [a,b] = a \rangle$
\bigskip
$
\chi_{pq}(\underbrace{a,\ldots, a}_r, \underbrace{b,\ldots,b}_s) =
\begin{cases}
1 & \text{if } p=r \text{ and } q=s \\
0 & \text{otherwise}
\end{cases}
$
\bigskip
$\Homol^n_{comm}(L,K) \simeq
\langle \chi_{pq} \vertbar p+q=n, p \text{ even}\rangle$
\bigskip
$
\chi_{pq} \smile \chi_{rs} =
\binom{p+r}{p} \, \binom{q+s}{q} \, \chi_{p+r, q+s}
$
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\begin{frame}{Zassenhaus}
$W_1^\prime(n) = \langle e_i = x^{(i+1)} \partial \vertbar -1 \le i \le 2^n-3 \rangle$
\bigskip
$\dim \Homol^2_{comm}(W_1^\prime(n),K) = n$
\bigskip
Basic cocycles:
$
e_i \vee e_j \mapsto \begin{cases}
1 &\text{if } i=j=2^k - 2, \text{or } \{i,j\} = \{-1,2^{k+1}-3\} \\
0 &\text{otherwise } ,
\end{cases}
$
$k = 0,\dots,{n-1}$.
\end{frame}
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\begin{frame}{Remaining questions}
\begin{itemize}
\item
Commutative cohomology as a derived functor?
\item
Compute commutative cohomology for various ``interesting'' algebras.
\item
An analog of the Hopf formula for the second degree \emph{homology}.
\item
Define the cup-product in the ``standard'' way.
\item
Algebras of cohomological dimension $1$?
\item
Euler-Poincar\'e characteristic?
\item
Whether the variety of commutative Lie algebras is Schreier?
\item
...
\end{itemize}
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References:
\begin{itemize}
\item V. Lopatkin and P. Zusmanovich, arXiv:1907.03690
\item F. Wagemann, arXiv:1908.06764
\end{itemize}
\vskip 2cm
\begin{center}
{\Huge That's all. Thank you.}
\end{center}
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