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\begin{document}
\title{Lecture 9: Homotopy Algebra}
\author{Alexander A. Voronov}
%\address{Department of Mathematics\\ M.I.T.,
%2-246\\ 77 Massachusetts Ave.\\ Cambridge, MA 02139-4307}
%\curraddr{}
%\email{voronov@math.mit.edu}
%\urladdr{http://www-math.mit.edu/~voronov/}
%\thanks{Research supported in part by an AMS Centennial Fellowship.}
%\subjclass{Primary 14H10; Secondary 32G15, 55P62}
\date{October 27, 1997}
\maketitle
\section{Homotopy Algebra}
The idea of a \emph{homotopy something algebra} is to relax the axioms
of the \emph{something algebra}, so that the usual identities are
satisfied up to homotopy. For example in a homotopy Lie algebra, the
Jacobi identity looks like
\[
[[a,b],c] \pm [[b,c],a] \pm [[c,a],b] \text{ is homotopic to zero}.
\]
Or in a homotopy Gerstenhaber (G-) algebra, the Leibniz rule is
\[
[a,bc] - [a,b]c \mp b[a,c] \text{ is homotopic to zero}.
\]
Usually, a homotopy something algebra arises when one wants to lift
the structure of a something algebra \emph{a priori} defined on \coh\
to the level of cochains.
\begin{xca}
Try to lift the BV structure on the \coh\ of a TVOA as defined by Lian
and Zuckerman, see Lecture 8 or \cite{lz}, to the level of cochains.
\end{xca}
This kind of relaxation seems to be too much for many, practical and
categorical, purposes, and one usually requires that the
null-homotopies, regarded as new operations, satisfy their own
identities, up to their own homotopy. These homotopies should also
satisfy certain identities up to homotopy and so on. This resembles
Hilbert's chains of syzygies in early homological algebra.
Operads are especially helpful when one needs to work with homotopy
something algebras. We already know that defining the class of
something algebras is equivalent to defining the something
operad. Thus, if we have an operad $\OO$, what is \emph{the} homotopy
$\OO$ operad? In such generality, there is no completely satisfactory
answer to this question. Here is one possible answer: \emph{a homotopy
$\OO$ operad} is a resolution $h\OO$ of $\OO$ in the category of
operads of complexes, \emph{i.e}., an operad of complexes whose \coh\
is $\OO[0]$, the operad $\OO$ sitting in degree zero, if it was an
operad of vector spaces, and the operad $\OO$ sitting in the original
degrees, if it was already an operad of graded vector spaces.
\begin{ex}
The singular chain operad $\{C_\bullet (D(n)) | n \ge 1\}$ of the
little disks operad $D$ is obviously a homotopy G-operad: its \coh\ is
the operad $H_\bullet (D)$, which we know is the G-operad. However,
this homotopy G-operad is too big. In an algebra over it, there are
even infinitely many dot products, corresponding to points in $D(2)$,
which are all homotopic to each other (because $D(2)$ is
connected). Also, the operad structure on the singular chains of a
topological operad is rather complicated: the map $C_\bullet (X)
\otimes C_\bullet (Y) \to C_\bullet (X \times Y)$ needed to define
the operad composition is defined using a cumbersome shuffle formula.
\end{ex}
However, one manages to define \emph{the} homotopy something operad
for certain specific classes of operads. For example, Ginzburg and
Kapranov \cite{gk} do it for so-called quadratic operads. Markl
\cite{markl} defines the homotopy something operad as the minimal
model, a notion he introduces, of the something operad. Since homotopy
something algebras are usually defined by concrete examples, we will
just describe a few examples of homotopy something operads.
\subsection{Homotopy Lie operad and algebras}
\begin{df}
A {\it homotopy Lie algebra} is a complex $V = \sum_{i \in \nz} V^i$
with a differential $d$, $d^2 = 0$, of degree 1 and a collection of
$n$-ary brackets:
\[
[v_1, \dots, v_n] \in V, \qquad v_1, \dots, v_n \in V,\; n \ge 2,
\]
which are homogeneous of degree $2-n$ and graded skew:
\[
[v_1, \dots, v_i, v_{i+1}, \dots , v_n] = - (-1)^{|v_i| |v_{i+1}|}
[v_1, \dots, v_{i+1}, v_i, \dots , v_n],
\]
$|v|$ denoting the degree of $v \in V$, and satisfy the relations
\begin{multline*}
\label{Q}
d[v_1, \dots, v_n] - (-1)^n \sum_{i=1}^n \epsilon(i) [v_1,
\dots, d v_i, \dots, v_n]
\\
= \sum_{\substack{k+l = n\\ k \ge 2, l \ge 1}}
\sum_{\substack{
\text{unshuffles } \sigma:\\
\{1,2, \dots, n\} = I_1 \cup I_2,\\
I_1 = \{i_1, \dots, i_k\}, \; I_2 = \{ j_1, \dots, j_{l}\}
}}
\epsilon (\sigma) \sign(\sigma)
(-1)^{kl} [[v_{i_1}, \dots, v_{i_k}], v_{j_1}, \dots, v_{j_{l}}],
\end{multline*}
where $\epsilon (i) = (-1)^{|v_1| + \dots + |v_{i-1}|}$ is the sign
picked up by taking $d$ through $v_1, \linebreak[0] \dots,
\linebreak[1] v_{i-1}$, $\sign(\sigma)$ is the sign of the permutation
$\sigma$, and $\epsilon (\sigma)$ is the sign picked up by the
elements $v_i$ passing through the $v_j$'s during the unshuffle of
$v_1, \dots , v_n$, as usual in superalgebra.
\end{df}
According to a Hinich-Schechtman theorem \cite{hs}, homotopy Lie
algebras can be described as algebras over a certain tree operad,
which is encoded in the topology of the moduli spaces due to Beilinson
and Ginzburg \cite{bg:1}. We will recall these results briefly.
\emph{To be finished\dots}
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