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% modified: pasha apr 18,28 2011
% modification: fixed a typo; correct location of slides; TeX cleanup
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\hyphenation{Ginz-burg Ka-pra-nov Ko-szu-li-ty al-geb-ras}
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\title{Non-Koszulity of the alternative operad and inversion of polynomials}
\author{Pasha Zusmanovich}
\date{April 19, 2011}
\begin{document}
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\begin{frame}
\titlepage
\begin{center}
based on joint work with Askar Dzhumadil'daev \\ \textsf{arXiv:0906.1272}
\end{center}
\end{frame}
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\begin{frame}{What is an operad?}
An \textit{operad} is a sequence $\mathcal P(n)$ of right $S_n$-modules equipped
with compositions
$$
\circ_i: \mathcal P(n) \times \mathcal P(m) \to \mathcal P(n+m-1)
$$
satisfying associativity-like conditions:
$$
(f \circ_i g) \circ_j h = f \circ_j (g \circ_{i-j+1} h)
$$
\begin{itemize}
\item
J. Stasheff, \emph{What is... an operad?}, Notices Amer. Math. Soc. June/July 2004.
\item
P. Cartier, \emph{What is an operad?},
The Independent Univ. of Moscow Seminars, 2005.
\end{itemize}
\uncover<2->{
\textbf{Primitive view: multilinear parts of relatively free algebras.}
$\mathcal P(n)$ = space of multilinear (nonassociative) polynomials of degree $n$.
}
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\begin{frame}{What is a Koszul operad?}
Koszulity = ``good'' homological properties.
Associative, Lie and associative commutative operads are Koszul.
\uncover<2->{
\begin{block}{Ginzburg--Kapranov criterion}
If a binary quadratic operad $\mathcal P$ over a field
of characteristic zero is Koszul, then
$$
g_{\mathcal P} (g_{\mathcal P^!} (t)) = t
$$
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 $\mathcal P$, and
$\mathcal P^!$ is the operad dual to $\mathcal P$.
\end{block}
}
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\begin{frame}{Examples of Poincar\'e series}
$$
g_{\mathcal Ass}(t) = \sum_{n=1}^\infty (-1)^n \frac{n!}{n!} t^n = - \frac{t}{1+t}
$$
$$
g_{\mathcal Comm}(t) = \sum_{n=1}^\infty (-1)^n \frac{1}{n!} t^n = e^{-t} - 1
$$
$$
g_{\mathcal Lie}(t) = \sum_{n=1}^\infty (-1)^n \frac{(n-1)!}{n!} t^n = -log(1+t)
$$
\end{frame}
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\begin{frame}{What is a dual operad?}
Pairing $\langle\,,\,\rangle$ on the space of multilinear (nonassociative) polynomials
of degree $3$:
$$
\Big\langle (x_i x_j) x_k, (x_{\sigma(i)} x_{\sigma(j)}) x_{\sigma(k)} \Big\rangle
= (-1)^\sigma
$$
$$
\Big\langle x_i (x_j x_k), x_{\sigma(i)} (x_{\sigma(j)} x_{\sigma(k)}) \Big\rangle
= -(-1)^\sigma
$$
$$
\Big\langle (x_i x_j) x_k, x_{i^\prime} (x_{j^\prime} x_{k^\prime}) \Big\rangle = 0
$$
$\sigma \in S_3$.
\bigskip
$R$ - relations in a binary quadratic operad
$R^!$ - dual space of relations under this pairing
\end{frame}
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\begin{frame}{Dual operads}
\begin{block}{Examples}
$\mathcal Ass^! = \mathcal Ass$
$\mathcal Lie^! = \mathcal Comm$
\end{block}
\uncover<2->{
\begin{block}{A remarkable fact}
If $A$, $B$ are algebras over operads dual to each other, then
$A \otimes B$ under the bracket
$$
[a \otimes b, a^\prime \otimes b^\prime] =
a a^\prime \otimes b b^\prime - a^\prime a \otimes b^\prime b
$$
for $a, a^\prime \in A$, $b, b^\prime \in B$, is a Lie algebra.
\end{block}
}
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\begin{frame}{Alternative algebras}
$$
(xy)y = x(yy)
$$
$$
(xx)y = x(xy)
$$
\bigskip
An example: the $8$-dimensional octonion algebra.
\uncover<2->{
\begin{block}{Theorem}
The alternative operad over a field of characteristic zero is not Koszul.
\end{block}
}
\uncover<3->{
\begin{block}{}
Proof by the Ginzburg--Kapranov criterion.
\end{block}
}
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\begin{frame}{Proof of the Theorem}
Easy part:
Dual alternative algebra: associative and $x^3 = 0$.
$$
g_{\mathcal Alt^!}(t) =
- t + t^2 - \frac{5}{6} t^3 + \frac{1}{2} t^4 - \frac{1}{8} t^5 .
$$
\uncover<2->{
Difficult part:
$$
g_{\mathcal Alt} (t) =
- t + t^2 - \frac{7}{6} t^3 + \frac{4}{3} t^4 - \frac{35}{24} t^5 + \frac{3}{2} t^6
+ O(t^7) .
$$
}
\uncover<3->{
Not Koszul by Ginzburg--Kapranov:
$$
g_{\mathcal Alt} (g_{\mathcal Alt^!} (t)) = t - \frac{11}{72} t^6 + O(t^7) .
$$
}
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\begin{frame}{Albert}
$\dim \mathcal Alt(n)$ for $n = 1, \dots, 6$
are computed with the help of \textit{Albert}
(developed in 1990s by David Pokrass Jacobs) and PARI/GP.
\bigskip
\texttt{http://justpasha.org/math/albert/ }
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\begin{frame}{Questions}
\begin{block}{Question}
Does the inverse of the polynomial
$$
g_{\mathcal Alt^!}(t) =
- t + t^2 - \frac{5}{6} t^3 + \frac{1}{2} t^4 - \frac{1}{8} t^5
$$
have alternating signs?
\end{block}
\uncover<2->{
\begin{block}{Another question (Martin Markl and Elizabeth Remm, 2009--2011)}
Does the inverse of the polynomial
$$
-t + t^8 - t^{15}
$$
have alternating signs?
\end{block}
}
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\begin{frame}{Three morals of this story}
\begin{itemize}
\item
One can do something in operads without really understanding them.
\item
Use open source. Make your software publically available.
\item
Questions about signs of inversions of polynomials are difficult. Study them!
\end{itemize}
\uncover<2->{
\bigskip
\begin{center}
{\Large That's all. Thank you.}
\end{center}
\bigskip
Slides at \texttt{http://justpasha.org/math/alternative/}
}
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