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\title{Mock-Lie algebras}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
XXVth International Conference on \\
Integrable Systems and Quantum Symmetries \\ 
Prague \\ June 2017
}


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\begin{frame}{Mock-Lie algebras}

By definition, mock-Lie algebras are ``commutative'' Lie algebras:
\begin{gather*}
xy = yx
\\
(xy)z + (zx)y + (yz)x = 0
\end{gather*}

\begin{block}{Fact 1}
Mock-Lie algebras are Jordan algebras.
\end{block}

\begin{block}{Proof}
Setting $z=x$ in the Jacobi identity gives $x^2 y = -2x(xy)$. Replacing
$x$ by $xy$ gives the Jordan identity $x^2(xy) = (x^2y)x$.
\end{block}

\begin{block}{Fact 2}
Mock-Lie algebras satisfy $x^3 = 0$.
\end{block}

\begin{block}{Proof}
Set $x=y=z$ in the Jacobi identity.
\end{block}

(All this assuming the characteristic of the base field $\ne 2,3$).

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\begin{frame}{Why bother?}

\begin{itemize}
\item Certain genetic algebras are mock-Lie;
\item Part of structure theory of Jordan algebras;
\item Appear in classification of certain cyclic operads;
\item 
It is curious to see what breaks in Lie algebras when anticommutativity
is replaced by commutativity;      
\item \textbf{Possibility of realization of physically-motivated Lie algebras};
\item \textbf{Possible relation to QYBE}.
\end{itemize}

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\begin{frame}{The dual life of mock-Lie algebras}

\begin{center}
{\small nil-index $3$ Jordan algebras}
$\begin{array}{l}
\includegraphics[scale=0.11]{yin_yang.png}
\end{array}$
{\small strange cousins of Lie algebras}
\end{center}

\bigskip

\begin{block}{Fact 3}
Finite-dimensional mock-Lie algebras are nilpotent.
\end{block}

\begin{block}{Proof}
Follows from the Jordan theory.
\end{block}

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\begin{frame}{Some history}

\begin{itemize}
\item 1960s-1970s Zhevlakov, Shestakov, et al.: \\
``Jordan algebras of nil-index $3$''
\item 1976 Abraham: some kind of genetic algebras 
\item 1995 Getzler--Kapranov: ``Mock-Lie algebras''
\item 1997 Okubo--Kamiya: ``Jordan-Lie algebras''
\item 1997 Hentzel--Jacobs--Sverchkov
\item 2014 Burde--Fialowski ``Jacobi-Jordan algebras''
\item 2015 Agore--Militaru ``Jacobi-Jordan algebras''
\item 2016 Gutierrez Fernandez--Garcia
\item 2017 Zusmanovich
\end{itemize}

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\begin{frame}{Examples}

\begin{block}{Example 1}
``Heisenberg'' algebra:
\begin{gather*}
\langle x_1, \dots, x_n, y_1, \dots, y_n, z \rangle  \\
x_i \cdot y_i = z
\end{gather*}
\end{block}

\begin{block}{Example 2}

Let $V$ be a vector space.
\begin{gather*}
V \oplus \bigwedge\nolimits^\varepsilon(V)  \\
(v,x) \cdot (w,y) = x \wedge w + y \wedge v,
\quad \text{where }v,w \in V, \> x,y \in \bigwedge\nolimits^\varepsilon(V)
\end{gather*}

\end{block}

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\begin{frame}{Representations}

(An old) Eilenberg's approach: consider $L \oplus V$ with square-zero $V$ and
an action of $L$ on $V$.
\begin{gather*}
\rho: L \to \End(V)
\\
\rho(xy)(v) = -\rho(x)\rho(y)v - \rho(y)\rho(x)v, 
\quad \text{where }x,y\in L, \> v\in V
\end{gather*}

In particular, the role of derivations is played by antiderivations:
$$
D(xy) = -D(x)y - xD(y) 
$$

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\begin{frame}{Cohomology}

\begin{block}{Fact 4 (Getzler--Kapranov)}
The mock-Lie operad is not Koszul.
\end{block}

\begin{block}{Proof}
The Koszul dual operad to mock-Lie is defined by
\begin{gather*}
xy = -yx \\
(xy)z = -x(yz)
\end{gather*}
Apply the Ginzburg--Kapranov criterion: 
$g_{\text{mock-Lie}}(g_{\text{dual mock-Lie}})(t) \ne t$
\end{block}

\begin{block}{Question 1}
Cohomology theory for mock-Lie algebras?
\end{block}

\bigskip

The standard ways to define low-degree cohomology -- via \emph{anti}derivations,
central extensions, deformations -- do not agree!

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\begin{frame}{``Dual mock-Lie''}

\begin{block}{Question 2}
Which ``interesting'' Lie algebras can be represented in the form
$(A \otimes B)^{(-)}$, where $A$ is a mock-Lie algebra, and $B$ is an
anticommutative antiassociative algebra? 
\end{block}

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\begin{frame}{Relation to QYBE}

\begin{block}{Fact 5 (Agore--Militaru)}
Let $A$ be a mock-Lie algebra, $z \in \Z(A)$, $\lambda \ne 0$ an element of the
ground field. Then the map $A \otimes A \to A \otimes A$ defined by
\begin{equation*}
a \otimes b \mapsto b \otimes a + \lambda z \otimes ab
\end{equation*}
is a solution of QYBE iff $A$ satisfies $(xy)z = 0$.
\end{block}

\begin{block}{Question 3}
Generalization to arbitrary mock-Lie algebras? To other kinds of *YBE
(QYBE with parameters, etc.)?
\end{block}

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\begin{frame}{Special and exceptional algebras}

\begin{block}{Fact 6 (Hentzel--Jacobs--Sverchkov)}
Mock-Lie algebras do not necessarily admit a faithful representation.
\end{block}

Equivalently:
\begin{itemize}
\item
in Lie parlance: the Ado theorem fails for mock-Lie;
\item
in Jordan parlance: there are exceptional mock-Lie algebras, i.e.
not every mock-Lie algebra can be embedded into an associative one
(with respect to anticommutator $a \circ b = \frac 12(ab + ba)$).
\end{itemize}

\begin{block}{Question 4}
Which ones do? 
Or: which mock-Lie algebras are special and which are exceptional?
\end{block}

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\begin{frame}{Special algebras}

\begin{block}{A (very) partial answer to Question 4}
Mock-Lie algebras of dimension $\le 6$ are special.
\end{block}

\begin{block}{1st proof}
Using Gr\"obner base computations in GAP/GBNP.
\end{block}

\begin{block}{2nd proof}
Using direct computations with the kernel of the canonical map $L \to U(L)$.
\end{block}

\begin{block}{3rd proof}
Using some Jordan theory.
\end{block}

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\begin{frame}{How to construct exceptional mock-Lie algebras}

For any of the special Jordan identities (Glennie, Shestakov, Medvedev, ...), 
construct a mock-Lie algebra not satisfying this identity using Albert 
(\url{http://www1.osu.cz/~zusmanovich/albert/}).

\medskip

A rigorous check that the given algebra does not satisfy the given identity in 
any characteristic $\ne 2,3$ can be done in GAP.

\medskip

The minimal dimension of so constructed algebra is $44$.

\begin{block}{Question 5 (a variation of Question 4)}
What is: \\
1) the minimal dimension \\
2) the minimal degree of nilpotency \\
of an exceptional mock-Lie algebra?
\end{block}

\begin{block}{A (very) partial answer}
1) between $7$ and $44$.

2) between $6$ and $9$.
\end{block}

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

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Slides at \url{http://www1.osu.cz/~zusmanovich/math.html}
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