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% file: covilha-2022.tex
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\documentclass{beamer}
\def\liebrack {\ensuremath{[\,\cdot\, , \cdot\,]}}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ker}{Ker}
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\title{On contact brackets on the tensor product}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
IV International Workshop on Non-Associative Algebras in Covilh\~{a}
\\
October 24, 2022
}
\begin{document}
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\begin{frame}{Poisson brackets}
$A$ associative commutative algebra with unit, with the skew-symmetric bracket
$\liebrack$
\begin{align*}
&[ab,c] = [a,c]b + [b,c]a \\
&[[a,b],c] + [[c,a],b] + [[b,c],a] = 0
\end{align*}
\begin{block}{A paradigmatic example}
$D,F$ derivations of $A$
$$
[a,b] = D(a)F(b) - F(a)D(b)
$$
\end{block}
\begin{block}{Fact}
For any Poisson brackets $\liebrack_A$ on $A$, and $\liebrack_B$ on $B$, there
is a Poisson bracket $\liebrack$ on $A \otimes B$ extending $\liebrack_A$ and
$\liebrack_B$.
\end{block}
\begin{block}{Proof}
$
[a \otimes b, a^\prime \otimes b^\prime] =
[a,a^\prime]_A \otimes bb^\prime +
aa^\prime \otimes [b,b^\prime]
$
\\
(tensor product of Poisson algebras).
\end{block}
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\begin{frame}{Contact brackets}
$A$ associative commutative algebra with unit, with the skew-symmetric bracket
$\liebrack$
\begin{align*}
&[ab,c] = [a,c]b + [b,c]a + [c,1]ab \\
&[[a,b],c] + [[c,a],b] + [[b,c],a] = 0
\end{align*}
\begin{block}{A paradigmatic example}
$D$ a derivation of $A$
$$
[a,b] = D(a)b - D(b)a
$$
\end{block}
\begin{block}{Question (Mart\'inez--Zelmanov 2019)}
Does for any Poisson bracket $\liebrack_A$ on $A$, and contact bracket
$\liebrack_B$ on $B$, there exist a contact bracket $\liebrack$ on
$A \otimes B$ extending $\liebrack_A$ and $\liebrack_B$?
\end{block}
\begin{block}{Answer}
Yes \& No.
\end{block}
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\begin{frame}{``No'' generally: a ``minimal'' example}
$A = K[x,y]/(x^2,y^2)$ with Poisson bracket $[x,y]_A = xy$
\bigskip
$B = K[x]/(x^2)$ with contact bracket $[1,x]_B = x$
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\begin{frame}{How to describe structures on the tensor product?}
Let one of $A$, $B$ is finite-dimensional. Then
$$
\Hom_K(A \otimes B \otimes A \otimes B) \simeq \Hom_K(A \otimes A,A) \otimes
\Hom_K(B \otimes B,B) ,
$$
so any contact bracket on $A \otimes B$ is of the form
$$
[a \otimes b, a^\prime \otimes b^\prime] =
\sum_i f_i(a,a^\prime) \otimes g_i(b,b^\prime)
$$
We deal with multiple conditions of the form $\sum_i S(f_i) \otimes T(g_i) = 0$
for some linear operators $S$ and $T$. For example, symmetrizing and
substituting $1$ in the ``contact condition'' for this bracket, we get
\begin{equation*}
\sum_{i}
\Big(f_i(a,a^{\prime\prime})a^\prime - f_i(a^\prime,a^{\prime\prime})a\Big)
\otimes
\Big(g_i(b, b^{\prime\prime})b^\prime - g_i(b^\prime,b^{\prime\prime})b\Big)
= 0 .
\end{equation*}
As $\Ker(S \otimes T) \simeq \Ker(S) \otimes \cdot + \cdot \otimes \Ker(T)$,
we may ``partition'' the set of indices $i$ such that either $S(f_i) = 0$, or
$T(g_i) = 0$.
The lemma on the next slide tells that we may do that simultaneously for two
such conditions.
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\begin{frame}{An old lemma and an application}
\begin{block}{An old lemma (Zusmanovich 2005)}
Let $U, W$ be two vector spaces,
$S, S^\prime \in \Hom(U, \>\cdot\>)$, $T, T^\prime \in \Hom(W, \>\cdot\>)$. Then
\begin{align*}
\Ker(S \otimes T) &\cap \Ker(S^\prime \otimes T^\prime) \\
\simeq\> &(\Ker S \cap \Ker S^\prime) \otimes W \\
+\> &\Ker S \otimes \Ker T^\prime \\
+\> &\Ker S^\prime \otimes \Ker T \\
+\> &U \otimes (\Ker T \cap \Ker T^\prime).
\end{align*}
\end{block}
\begin{block}{An example of application}
$$
\mathcal K^-\big(A \otimes K[x]/(x^2)\big) \simeq
\mathcal K^-(A) \oplus \mathcal K^-(A) \oplus \Der(A) \oplus A ,
$$
where $\mathcal K^-(A)$ is the space of skew-symmetric linear maps on $A$ with the
``contact condition''.
\end{block}
\end{frame}
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\begin{frame}{``Yes'' for polynomial(-like) algebras}
For some algebras, each contact bracket is the sum of paradigmatic examples,
i.e., is of the form
$$
\sum_i \Big(D_i(a) F_i(b) - D_i(b) F_i(a)\Big) + D(a)b - D(b)a .
$$
Examples: $K[x_1,\dots,x_n]$ and $K[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$.
\bigskip
For such algebras $A$ and $B$, any contact brackets on $A$ and $B$ are extended
to a contact bracket on $A \otimes B$.
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{\small
The text is available as
\texttt{https://web.osu.cz/$\sim$Zusmanovich/papers/tensprodcontact.pdf}
}
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\begin{center}
{\Huge That's all. Thank you.}
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