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\begin{document}
\title{Non-koszulity of the alternative operad and inversion of polynomials \\
(Extended abstract)}
\author{Pasha Zusmanovich}
\address{
Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5,
Tallinn 19086, Estonia
}
\email{pasha.zusmanovich@ttu.ee}
\date{March 25, 2011}
\maketitle
Operads are in fashion nowadays. \textit{Koszulity} is an important property of operads,
saying, roughly, that given operad possesses nice homological properties. Operads
governing the behavior of all major classes of algebras -- associative, Lie, Poisson,
etc. -- are Koszul, what explains, retroactively, why these classes of algebras have
nice and reach (co)homological theories.
It is naturally a popular occupation these days to establish whether a given operad
is Koszul or not.
In doing so, one may use a \textit{Ginzburg--Kapranov criterion} that says
that if an operad $\mathcal P$ defined over a field of characteristic zero
and satisfying some natural restrictions, is Koszul, then
\begin{equation}\label{gk}
g_{\mathcal P} (g_{\mathcal P^!} (t)) = t ,
\end{equation}
where
$$
g_{\mathcal P}(t) = \sum_{n=1}^\infty (-1)^n \frac{\dim \mathcal P(n)}{n!} t^n
$$
is the Poincar\'e series of the operad $\mathcal P$, and $\mathcal P^!$
is the Koszul dual of $\mathcal P$.
So, if we are (un)lucky, and the equality (\ref{gk}) does not hold for an operad
$\mathcal P$, then $\mathcal P$ is not Koszul.
We are using this approach to establish a non-Koszulity of the operad $\mathcal Alt$
governing the behavior of alternative algebras, answering a question of Loday.
\textit{Alternative algebras} are generalization of associative algebras and are
determined by two identities of ``weak associativity'':
\begin{align*}
(xy)y &= x(yy) \\
(xx)y &= x(xy) .
\end{align*}
An example of a non-associative alternative algebra is the famous octonion
algebra.
Some authors expressed a viewpoint that non-Koszulity is a
rather pathological property, and all ``occuring in the real life''
algebras should be algebras over a Koszul operad.
As we see, alternative algebras provide a ``real life'' example
violating this principle.
So, first, we need to compute identities defining an operad dual to the alternative one.
This is a matter of a simple exercise, and the corresponding identities are, first,
associativity, and, second, $x^3 = 0$.
Next we should compute enough first terms in the dimension sequence
$\dim\mathcal P(n)$ for the alternative and dual alternative operads, to achieve the
violation of (\ref{gk}). These terms are nothing
but dimensions of the multilinear components of the free algebras of rank $n$
in the corresponding varieties of algebras. So, from now on, we can forget about operads
and think in much more pedestrian terms.
For the dual alternative algebras, i.e. for associative algebras satisfying the identity
$x^3 = 0$, this is easy: the corresponding Poincar\'e series is actually a polynomial
\begin{equation}\label{altbang}
g_{\mathcal Alt^!} (t)
= - t + t^2 - \frac{5}{6} t^3 + \frac{1}{2} t^4 - \frac{1}{8} t^5 .
\end{equation}
Free alternative algebras are much more difficult objects than,
for example, their associative or Lie counterparts, and are still not understood
sufficiently well, and to compute the corresponding dimension sequence is much more
difficult. For this, we utilize a program \textit{Albert} written at the beginning of
1990s by the team from Clemson University lead by David Pokrass Jacobs.
This is a remarkably useful program for dealing with identities of algebras,
and, in particular, for computations of the corresponding dimensions.
A modified version of this program, suited for our needs, is located at
\texttt{http://justpasha.org/math/albert/} .
Computations of such things over the field of rational numbers, due to explosive
growth of numerators and denominators occuring at intermediate stages of
computations, seem to be unfeasible, and Albert computes over prime fields.
Juggling with Chinese-remainder-type arguments, we produce the number of primes
sufficient to deduce the result in characteristic zero, and perform computations
for all these primes. The beginning terms of the corresponding Poincar\'e series are
$$
g_{\mathcal Alt} (t)
=
- t + t^2 - \frac{7}{6} t^3 + \frac{4}{3} t^4 - \frac{35}{24} t^5 + \frac{3}{2} t^6
+ O(t^7) ,
$$
and
$$
g_{\mathcal Alt} (g_{\mathcal Alt^!} (t)) = t - \frac{11}{72} t^6 + O(t^7) ,
$$
what violates (\ref{gk}).
One may try to avoid difficult computations of $g_{\mathcal Alt} (t)$, and argue as
follows. Look at the inverse of polynomial (\ref{altbang}). If, at certain place,
the consecutive terms of this inverse will have the same sign, then this inverse cannot
be a Poincar\'e series of any operad, and we get contradiction with (\ref{gk}).
Interestingly enough, this does not seem to be the case for up to the first
$1000$ terms: the signs are really alternating.
\begin{question}
Does the inverse of the polynomial (\ref{altbang}) have alternating signs?
If yes, what combinatorial interpretation this may have?
\end{question}
The same kind of questions arise in similar contexts, for example in \cite{markl-remm}
the same is asked about the polynomial $-t + t^8 - t^{15}$.
Such questions seem to be, somewhat surprisingly, difficult.
\medskip
Based on the work with Askar Dzhumadil'daev \cite{dz}.
\begin{thebibliography}{MR}
\bibitem[DZ]{dz} A. Dzhumadil'daev and P. Zusmanovich,
\emph{The alternative operad is not Koszul}, Experiment. Math., to appear;
\textsf{arXiv:0906.1272}
(a more recent version may be available at \texttt{http://justpasha.org/math/ }).
\bibitem[MR]{markl-remm} M. Markl and E. Remm,
\emph{Operads for $n$-ary algebras -- calculations and conjectures},
\textsf{arXiv:1103.3956v1}.
\end{thebibliography}
\end{document}
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