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\newtheorem*{lemma}{Lemma}
\def\liebrack {{\ensuremath{[\,\cdot\, , \cdot\,]}}}
\DeclareMathOperator{\Ker}{Ker}
\begin{document}
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%% Please use the environment "talk" for each abstract.
%% It has three obligatory and one optional argument. The syntax is:
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%% \begin{talk}[coauthors]{Firstname-of-speaker Lastname-of-speaker}{Title of the talk}{Author Sorting Index}
%% .....
%% \end{talk}
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%% their "non-special" version, eg replace \"a by a, \'a by a, etc.
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\begin{talk}{Pasha Zusmanovich}
{Contact brackets and other structures on the tensor product}
{Zusmanovich, Tensor products}
\noindent
The purpose of this report is once more to call attention to an elementary and,
in some cases, very effective technique of computing various kinds of structures
on tensor products. Such problems often can be reduced to the simultaneous
evaluation of kernels of several tensor product maps, i.e., maps of the
form $S \otimes T$, where $S$ and $T$ are linear operators on the respective
spaces of linear maps. Using the fact that
\begin{equation}\label{eq-ker}\tag{$\bigstar$}
\Ker(S \otimes T) = \Ker(S) \otimes * + * \otimes \Ker(S) ,
\end{equation}
the question reduces to evaluation of the intersection of several linear spaces
having the form as on the right-hand side of (\ref{eq-ker}), for various
operators $S$ and $T$. The intersection of two such spaces satisfies the
distributivity, and so can be handled effectively, due to the following
elementary linear algebraic lemma:
\begin{lemma}[Lemma 1.1 in \cite{low}]
Let $U_1, U_2$ be subspaces of a vector space $U$, and $V_1, V_2$ be subspaces
of a vector space $V$. Then
$$
(U_1 \otimes V + U \otimes V_1) \cap (U_2 \otimes V + U \otimes V_2) =
(U_1 \cap U_2) \otimes V + U_1 \otimes V_2 + U_2 \otimes V_1 +
U \otimes (V_1 \cap V_2) .
$$
\end{lemma}
This technique was used for the first time in \cite{low} to
derive some formulas for the low degree cohomology of current Lie algebras,
i.e., Lie algebras of the form $L \otimes A$, where $L$ is a Lie algebra, and
$A$ is an associative commutative algebra. The paper \cite{compendium} contains
further results about such cohomology, as well as about Poisson and Hom-Lie
structures on current and related Lie algebras. The last our result in this
direction is in \cite{contact}, which answers a recent question from
\cite{martinez-zelm} about extension of contact bracket on the tensor product
from the bracket on the factors.
Recall that the contact bracket on a commutative associative algebra $A$ with
unit is a bilinear map $\liebrack: A \times A \to A$ such that
$$
[ab,c] = [a,c]b + [b,c]a + [c,1]ab
$$
for any $a,b,c \in A$. Contact brackets are an obvious generalization of Poisson
brackets, the latter being contact brackets satisfying $[A,1] = 0$. It was asked
in \cite{martinez-zelm} whether, given contact brackets on two algebras $A$ and
$B$, is it always possible to extend them to the tensor product $A \otimes B$?
In \cite{contact}, using some general formulas for the space of contact brackets
on some particular classes of algebras, a procedure was devised for constructing
examples showing that such extension is not always possible.
This linear algebraic method is sometimes very effective, but its applicability
is severely limited by the fact that no statement similar to Lemma is true for
intersection of three or more spaces. The proper contexts of Lemma might be
criteria for distributivity of a set of subspaces of a vector space (for an exposition, see, for example,
\cite[Chap. 1, \S 7]{pp}) and, more speculatively, the ``four
subspaces problem'' of Gelfand--Ponomarev, \cite{gelfand-ponomarev-intro}.
\begin{thebibliography}{99}
\bibitem{gelfand-ponomarev-intro} I.M. Gelfand and V.A. Ponomarev,
\emph{Problems of linear algebra and classification of quadruples of subspaces
in a finite-dimensional vector space},
Hilbert Space Operators and Operator Algebras,
Colloq. Math. Soc. J\'anos Bolyai \textbf{5} (1972), 163--237.
\bibitem{martinez-zelm} C. Mart\'inez and E. Zelmanov,
\emph{Brackets, superalgebras and spectral gap},
S\~ao Paulo J. Math. Sci. \textbf{13} (2019), no.1, 112--132.
\bibitem{pp} A. Polishchuk and L. Positselski,
\emph{Quadratic Algebras}, AMS, 2005.
\bibitem{low}
P. Zusmanovich,
\emph{Low-dimensional cohomology of current Lie algebras and analogs of the
Riemann tensor for loop manifolds},
Lin. Algebra Appl. \textbf{407} (2005), 71--104.
\bibitem{compendium}
\bysame, \emph{A compendium of Lie structures on tensor products},
J. Math. Sci. \textbf{199} (2014), no.3, 266--288.
\bibitem{contact}
\bysame, \emph{On contact brackets on the tensor product},
Lin. Multilin. Algebra \textbf{70} (2022), no.19, 4695--4706.
\end{thebibliography}
\end{talk}
\end{document}