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\begin{document}

\title{Lecture 6: Operads via Generators and Relations}
\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 9, 1997}

\maketitle

\section{Operads via Generators and Relations}

The tree operads that we looked at in Lecture 4, such as the
associative and the Lie operads, are actually operads defined by
generators and relations. Here is a way to define such operads in
general. To fix notation, assume throughout this section that we work
with operads $\OO(n)$, $ n \ge 1$, of vector spaces.

\begin{df}
An \emph{ideal} in an operad $\OO$ is a collection $\II$ of
$S_n$-invariant subspaces $\II(n) \subset \OO(n)$, for each $n \ge 1$,
such that whenever $i \in \II$, $\gamma(\dots, i, \dots) \in \II$.
\end{df}

The intersection of an arbitrary number of ideals in an operad is also
an ideal, and one can define the ideal generated by a subset in $\OO$
as the minimal ideal containing the subset.

\begin{df}
The \emph{quotient operad} $\OO/\II$ is the collection $\OO(n)/\II(n)$,
$n \ge 1$, with the structure of operad induced by that on $\OO$.
\end{df}

The \emph{free operad $F(V)$ generated by a collection $V = \{ V(n) \;
| \; n \ge 1 \}$ of vector spaces}, is defined as follows.
\[
F(V)(n) = \bigoplus_{\text{$n$-trees $T$}} V(T),
\]
where the summation runs over all planar rooted trees $T$ with $n$
labeled leaves and
\[
V(T) = \bigotimes_{v \in T} V(\In(v)),
\]
the summation running over all vertices $v$ of the tree $T$, $\In(v)$
being the number of incoming edges for the vertex $v$ (the edges are
directed toward the root). In other words, an element of $F(V)(n)$ is
an planar $n$-tree whose vertices decorated with elements of
$V$. There is a special tree with no vertices:

\centerline{\epsfig{file=id.eps,height=0.5in}}

\noindent
The component $F(V)(1)$ contains the one-dimensional subspace spanned
by this tree.

The following data defines an operad structure on $F(V)$.

\begin{enumerate}

\item The identity element is the special tree in $F(V)(1)$ with no 
vertices.

\item The symmetric group $S_n$ acts on $F(V)(n)$ by relabeling the inputs.

\item The operad composition is given by grafting the roots of trees
to the leaves of another tree. No new vertices are created.

\end{enumerate}

\begin{df}
Now let $R$ be a subset of $F(V)$, i.e., a collection of subsets $R(n)
\subset F(V)(n)$. Let $(R)$ be the ideal in $F(V)$ generated by
$R$. The quotient operad $F(V)/(R)$ is called the \emph{operad
generated by $V$ with defining relations $R$}.
\end{df}

\begin{xca}
The associative operad of Lecture 4 is the operad generated by $V =
V(2) = k$ with defining relation given by the associativity condition,
see Section 1.2.2 of Lecture 4.
\end{xca}

\begin{xca}
The Lie operad of Lecture 4 is the operad generated by $V = V(2) = k$
with defining relations given by the skew symmetry condition and the
Jacobi identity, see Section 1.2.3 of Lecture 4.
\end{xca}

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