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\title{Non-semigroup gradings of associative algebras}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{
The First International Workshop \\ Non-associative algebras in C\'adiz \\
February 2018
}
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Based on:
\begin{itemize}
\item Lin. Algebra Appl. \textbf{523} (2017), 52--58 = arXiv:1609.03924
\item J. Algebra \textbf{324} (2010), 3470--3486 = arXiv:0907.2034
\end{itemize}
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These slides are at \texttt{http://www1.osu.cz/$\sim$zusmanovich/}
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\begin{frame}{Do non-semigroup gradings exist?}
$$
A = \bigoplus_{\alpha \in G} A_{\alpha}
$$
An algebra graded by a set $G$ with a (partial) binary operation $*$:
$$
A_\alpha A_\beta \subseteq A_{\alpha * \beta} \text{ if } A_\alpha A_\beta \ne 0
$$
\begin{block}{Question (vague)}
How identities of $A$ and $G$ are related?
\end{block}
\begin{block}{Fact}
If $A$ is commutative or anticommutative, then $G$ can be embedded into a
commutative magma.
\end{block}
\begin{block}{Question (more concrete)}
Is it true that if $A$ is associative or Lie, then $G$ can be embedded into a
semigroup?
\end{block}
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\begin{frame}{Do non-semigroup gradings exist? Part II}
\begin{block}{Answer (for Lie algebras)}
Yes, it can (Patera and Zassenhaus 1989).
\end{block}
\bigskip
\uncover<2->{
\textbf{Turned out to be wrong $\sim$20 years later:}
There are Lie algebras with a non-semigroup grading! (Elduque 2006,2009).
\begin{block}{Example}
$L = \langle a,u \rangle \oplus \langle v \rangle \oplus \langle w \rangle$
$[a,u\>] = u$
$[a,v\>] = w$
$[a,w] = v$
\end{block}
\begin{block}{Question (remaining)}
What about (classical) simple Lie algebras?
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}
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\begin{frame}{Do non-semigroup gradings exist? Part III}
\begin{block}{Answer (for associative algebras)}
There are associative algebras with a non-semigroup grading (Zusmanovich 2017).
\end{block}
\bigskip
(The smallest known such algebra is $6$-dimensional).
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\begin{frame}{A method to construct non-semigroup gradings}
Take $\delta$-derivation $D$ of an algebra $A$, i.e.
$$
D(ab) = \delta \Big(D(a)b + aD(b)\Big)
$$
and consider the root space decomposition
$$
A = \bigoplus A_\lambda
$$
with respect to $D$. Then
$$
A_\lambda A_\mu \subseteq A_{\delta (\lambda + \mu)} .
$$
The ``multiplication''
$$
\lambda * \mu = \delta(\lambda + \mu)
$$
is nonassociative (subject to some conditions on $\delta$).
\bigskip
All non-semigroup gradings constructed so far follow this scheme.
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\begin{frame}{More questions}
\begin{enumerate}
\item
What is the smallest dimension of an associative algebra with a non-semigroup
grading?
\item
It is true that any grading of a full matrix algebra is a semigroup grading?
\item
Given a grading of a Lie algebra, is it possible to construct a grading of its
(restricted) universal enveloping algebra?
\item
Does the presence/absence of non-semigroup gradings of algebras over a binary
quadratic operad $\mathscr P$ entails the same for algebras over the operad
Koszul dual to $\mathscr P$?
\end{enumerate}
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{\Huge
That's all. Thank you.
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