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\title{CD algebras}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
``Non-associative algebras in M\'alaga'' \\
February 7, 2020
}
\begin{document}
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\begin{frame}
\titlepage
\begin{center}
(Joint work with Ivan Kaygorodov)
\end{center}
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\begin{frame}{CD algebras}
CD algebras are anticommutative algebras such that
$[R_a,R_b]$ is a derivation for any two elements $a,b$ of the algebra.
\bigskip
Such commutative algebras = Lie triple algebras
\\
(Bertram, Dzhumadil'daev, Jordan--R\"uhaak, Osborn, Sidorov, ..., 1969--2018)
\bigskip
What about \emph{anti}commutative ones?
\end{frame}
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\begin{frame}{Known classes of anticommutative algebras}
$$
\begin{array}{rcl}
&& \text{Malcev} \>\subset\> \text{ Binary Lie} \\
& \rotatebox{45}{$\subset$} & \\
\text{Lie} & \\
& \rotatebox{-45}{$\subset$} & \\
& & \text{Sagle}
\end{array}
$$
\vskip 20pt
binary Lie: $J([x,y],x,y) = 0$
\smallskip
Malcev: $J(x,y,[x,z]) = [J(x,y,z),x]$
\smallskip
Sagle: $[J(x,y,z),t] = J(t,z,[x,y]) + J(t,y,[z,x]) + J(t,x,[y,z])$
\end{frame}
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\begin{frame}{How CD fits into the picture?}
\begin{equation*}
\begin{array}{rcl}
&& \text{Binary Lie} \\
& \rotatebox{45}{$\subset$} \\
\text{Lie} \>\subset\> \text{Malcev} \cap \text{Sagle} \>\subset\> \text{CD} \\
& \rotatebox{-45}{$\subset$} \\
&& \text{Almost Lie}
\end{array}
\end{equation*}
\bigskip
\begin{equation*}
\begin{gathered}
\text{Malcev} \>\cap\> \text{CD} =
\text{Sagle} \>\cap\> \text{CD} =
\text{Malcev} \>\cap\> \text{Sagle}
\\
\text{Binary Lie} \>\cap\> \text{Almost Lie} = \text{CD}
\end{gathered}
\end{equation*}
\vskip 20pt
Almost Lie: $[J(x,y,z),t] = 0$
\end{frame}
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\begin{frame}{CD algebras are ``central extensions'' of Lie algebras}
\begin{block}{Corollary 1 (to inclusion at the previous slide)}
For any CD algebra $A$, $A/Z(A)$ is a Lie algebra.
\end{block}
\begin{block}{Corollary 2}
Any simple CD algebra is a Lie algebra.
\end{block}
\begin{block}{An old open question}
Whether any simple binary Lie algebra is Malcev?
\end{block}
\textbf{Yes} in the class of finite-dimensional algebras over a field of
characteristic $0$ (Grishkov 1980).
\textbf{Yes} in the class of CD algebras.
\end{frame}
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\begin{frame}{Cohomology? Variant 1}
\textbf{Central extensions + derivations + deformations}
$$
0 \to LZ(M) \oplus (A \otimes M) \overset{\dcobound^0}\longrightarrow C^1(A,M)
\overset{\dcobound^1}\longrightarrow C^2(A,M)
\overset{\dcobound^2}\longrightarrow C^4(A,M)
$$
$LZ(M) = \set{m \in M}
{xy \bullet m - x \bullet (y \bullet m) + y \bullet (x \bullet m) = 0
\forall x,y \in M}$, the ``Lie center''
$C^n(A,M)$ - skew-symmetric multilinear maps
\medskip
$\dcobound^0(m)(b) = b \bullet m$
$\dcobound^0(a \otimes m)(b) = a \bullet (b \bullet m) - b \bullet (a \bullet m)$
$
\dcobound^1(\varphi)(x,y) =
\varphi(xy) - x \bullet \varphi(y) + y\bullet \varphi(x)
$
$
\dcobound^2(\varphi)(x,y,a,b) =
\varphi((xy)a,b) - \varphi((xy)b,a)
- \varphi((xa)b,y) + \varphi((xb)a,y) + \varphi((ya)b,x) - \varphi((yb)a,x)
+ a \bullet \varphi(xy,b) - b \bullet \varphi(xy,a)
- x \bullet \varphi(ya,b) + x \bullet \varphi(yb,a)
+ y \bullet \varphi(xa,b) - y \bullet \varphi(xb,a)
- a \bullet (b \bullet \varphi(x,y)) + b \bullet (a \bullet \varphi(x,y))
- x \bullet (a \bullet \varphi(y,b)) + x \bullet (b \bullet \varphi(y,a))
- y \bullet (b \bullet \varphi(x,a)) + y \bullet (a \bullet \varphi(x,b))
$
\end{frame}
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\begin{frame}{Cohomology? Variant 2}
\textbf{``CD derivations''}
$$
0 \to M \overset{\dcobound^0}\longrightarrow C^1(A,M)
\overset{\dcobound^1}\longrightarrow C^3(A,M)
$$
$\dcobound^0(m)(x) = x \bullet m$
$\dcobound^1(\varphi)(x,y,a) =
\varphi((xy)a) - a \bullet \varphi(xy)
- y \bullet \varphi(xa) + x \bullet \varphi(ya)
+ y \bullet (a \bullet \varphi(x)) - x \bullet (a \bullet \varphi(y))
$
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\begin{frame}{Cohomology? Variant 3}
\textbf{By analogy with the Chevalley--Eilenberg}
\begin{gather*}
C^1(A,M) \overset{\dcobound}{\to} C^3(A,M) \overset{\dcobound}{\to}
C^5(A,M) \overset{\dcobound}{\to} \dots
\\
C^2(A,M) \overset{\dcobound}{\to} C^4(A,M) \overset{\dcobound}{\to}
C^6(A,M) \overset{\dcobound}{\to} \dots
\end{gather*}
{\tiny
\begin{multline*}
\dcobound(\varphi)(x,y,a_1, \dots, a_n)
\\ =
\sum_{i=1}^n (-1)^i \Big(
\varphi((xy)a_i, a_1, \dots, \widehat{a_i}, \dots, a_n)
+ a_i \bullet \varphi(xy, a_1, \dots, \widehat{a_i}, \dots, a_n)
\\
- x \bullet \varphi(ya_i, a_1, \dots, \widehat{a_i}, \dots, a_n)
+ y \bullet \varphi(xa_i, a_1, \dots, \widehat{a_i}, \dots, a_n)
\\
- x \bullet (a_i \bullet \varphi(y, a_1, \dots, \widehat{a_i}, \dots, a_n))
+ y \bullet (a_i \bullet \varphi(x, a_1, \dots, \widehat{a_i}, \dots, a_n))
\Big)
\\ + \sum_{1 \le i < j \le n} (-1)^{i+j+n+1} \Big(
\varphi(
(xa_i)a_j, y, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n)
- \varphi(
(xa_j)a_i, y, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n)
\\
- \varphi(
(ya_i)a_j, x, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n)
+ \varphi(
(ya_j)a_i, x, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n)
\\
+ a_i \bullet (a_j \bullet
\varphi(x, y, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n))
- a_j \bullet (a_i \bullet
\varphi(x, y, a_1, \dots, \widehat{a_i}, \dots, \widehat{a_j}, \dots, a_n))
\Big)
\end{multline*}
}
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\begin{frame}{Second CD cohomology}
For any Lie algebra $L$ and an $L$-module $M$,
$$
\Homol^2(L,M) \subseteq \Homol_{CD}^2(L,M) .
$$
\begin{block}{Theorem (``Second CD Whitehead lemma'')}
($p=0$) For any simple finite-dimensional Lie algebra $L$, and any
finite-dimensional $L$-module $M$,
$$
\Homol_{CD}^2(L,M) = 0 .
$$
\end{block}
\begin{block}{Conjecture}
($p \ne 2,3$) For any simple finite-dimensional Lie algebra $L$,
$$
\Homol_{CD}^2(L,K) = \Homol^2(L,K) .
$$
\end{block}
Supported by computations in GAP.
\end{frame}
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\begin{frame}{A new invariant in characteristic $2$}
For some simple Lie algebras in the list of simple Lie algebras of dimension
$\le 20$ (Eick 2010), $\Homol_{CD}^2(L,K) \ne \Homol^2(L,K)$, and $\Homol_{CD}^2(L,K)$ distinguishes
some simple Lie algebras not distinguishable by the ``classical'' invariants.
\end{frame}
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\begin{frame}{Further questions}
\textbf{1)}
How ``far'' a CD algebra can be from Lie algebras? Describe
CD algebras $A$ with $LZ(A) = Z(A)$.
\bigskip
\textbf{2)}
Study free CD algebras. Are they central extensions of free Lie algebras?
\bigskip
\textbf{3)} Study ``CD speciality''.
\bigskip
\textbf{4)}
Study representations of CD algebras. An analog of the Ado theorem?
\bigskip
\textbf{5)}
CD algebras without conditions of commutativity or anticommutativity?
\end{frame}
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\begin{center}
{\Huge That's all. Thank you.}
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