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\section*{pre-Lie algebra}

\hypertarget{prelie_algebras}{}\section*{{Pre-Lie algebras}}\label{prelie_algebras}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{relation_to_the_work_of_connes_and_kreimer}{Relation to the work of Connes and Kreimer}\dotfill \pageref*{relation_to_the_work_of_connes_and_kreimer} \linebreak
\noindent\hyperlink{selfreferentiality}{Self-referentiality}\dotfill \pageref*{selfreferentiality} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{pre-Lie algebra} is a [[vector space]] $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that

\begin{equation}
[L_a, L_b] = L_{[a,b]}
\label{basic_identity}\end{equation}
for all $a, b\in A$. Here $L_a$ is the operation of left multiplication by $a$:

\begin{displaymath}
L_a b = a \cdot b
\end{displaymath}
and $[L_a, L_b]$ is the usual commutator of operators using [[composition]]:

\begin{displaymath}
[L_a, L_b] = L_a L_b - L_b L_a
\end{displaymath}
while $[a,b]$ is the commutator defined using the $\cdot$ operation:

\begin{equation}
[a,b] = a \cdot b - b \cdot a
\label{torsion_free_identity}\end{equation}
Unravelling this, we see a pre-Lie algebra is vector space $A$ equipped with a bilinear operation $\cdot: A \times A \to A$ such that

\begin{equation}
a \cdot (b \cdot c) - b \cdot (a \cdot c) = 
(a \cdot b) \cdot c - (b \cdot a) \cdot c
\label{identity}\end{equation}
More precisely, this is a \textbf{left} pre-Lie algebra. We can also define right pre-Lie algebras.

Every [[associative algebra]] is a pre-Lie algebra, but not conversely. The reason pre-Lie algebras have the name they do is that this weakening of the concept of associative algebra is still enough to give a [[Lie algebra]]! In other words: it is well-known that if $A$ is an associative algebra, the operation

\begin{displaymath}
[a,b] = a \cdot b - b \cdot a
\end{displaymath}
makes $A$ into a Lie algebra. But this is also true for pre-Lie algebras! It is a fun exercise to derive the [[Jacobi identity]] from equation \eqref{identity}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

First, given a [[manifold]] with a flat torsion-free [[connection]] $\nabla$ on its [[tangent bundle]], we can make the space of [[tangent vector fields]] into a pre-Lie algebra by defining

\begin{displaymath}
v \cdot w = \nabla_v w
\end{displaymath}
The definition of `flat' is precisely \eqref{basic_identity}, whereas that of `torsion-free' is precisely \eqref{torsion_free_identity}. The Lie algebra arising from this pre-Lie algebra is just the usual Lie algebra of vector fields.

Second, suppose $O$ is a [[linear operad]], and let $A$ be the free $O$-[[algebra of an operad|algebra]] on one generator. As a vector space we have

\begin{displaymath}
A = \bigoplus_{n} O_n/S_n \, .
\end{displaymath}
Here $S_n$ is the [[symmetric group]], which acts on the space $O_n$ of $n$-ary operations of $O$. Moreover, $A$ becomes a pre-Lie algebra in a manner described here:

\begin{itemize}%
\item Dominique Manchon, \href{http://math.univ-bpclermont.fr/~manchon/biblio/ESI-prelie2009.pdf}{A short survey on pre-Lie algebras}

\end{itemize}
Third, the [[Hochschild cohomology|Hochschild]] [[chain complex]] of any associative algebra, with grading shifted down by one, can be given the structure of a `[[graded object|graded]] pre-Lie algebra', as discovered by Gerstenhaber and described here:

\begin{itemize}%
\item Justin Thomas, \href{http://www.math.northwestern.edu/~jdthomas/Talk%20Notes/Hoch%20Cohomology.pdf}{Graduate student seminar: Hochschild cohomology}

\end{itemize}
In fact it was Gerstenhaber who coined the term `pre-Lie algebra', for this reason.

\hypertarget{relation_to_the_work_of_connes_and_kreimer}{}\subsection*{{Relation to the work of Connes and Kreimer}}\label{relation_to_the_work_of_connes_and_kreimer}

Connes and Kreimer formalized the process of \href{http://ncatlab.org/nlab/show/renormalization#hopfalgebraic_renormalization_3}{renormalization} using a certain [[Hopf algebra]] built from [[Feynman diagram]]s. More abstractly we can understand the essence of their construction using a Hopf algebra built from rooted trees, as explained here:

\begin{itemize}%
\item John Baez, This Week's Finds in Mathematical Physics, Week 299. (\href{http://math.ucr.edu/home/baez/week299.html}{web}) (\href{http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html}{blog})

\end{itemize}
The key is to form the free pre-Lie algebra on one generator, then turn this into a Lie algebra as described above, and then form the universal enveloping of that, which is a cocommutative Hopf algebra. Finally, the restricted dual of this cocommutative Hopf algebra is the commutative Hopf algebra considered by Connes and Kreimer here:

\begin{itemize}%
\item Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, \emph{Commun. Math. Phys.} \textbf{199} (1998), 203--242 (\href{http://arxiv.org/abs/hep-th/9808042}{arXiv})

\end{itemize}
Pre-Lie algebras are algebras of a [[linear operad]] called $PL$. The space $PL_n$ has a basis given by labelled rooted [[trees]] with $n$ vertices, and the $i$th partial composite $s \circ_i t$ is given by summing all the possible ways of inserting the tree $t$ inside the tree $s$ at the vertex labelled $i$. For details see:

\begin{itemize}%
\item Fr\'e{}d\'e{}ric Chapoton, Muriel Livernet, Pre-Lie algebras and the rooted trees operad, \emph{Int. Math. Res. Not.} 2001 (2001), 395-408.

\end{itemize}
The free pre-Lie algebra on one generator is thus

\begin{displaymath}
\bigoplus_{n} PL_n /S_n  \,
\end{displaymath}
so the description of $PL_n$ in terms of rooted trees gives a kind of `explanation' of the relation between the Connes--Kreimer Hopf algebra and rooted trees.

\hypertarget{selfreferentiality}{}\subsection*{{Self-referentiality}}\label{selfreferentiality}

Pre-Lie algebras have a strange self-referential feature. Every operad of a large class gives a pre-Lie algebra, but the operad for pre-Lie algebras is one of this class! This raises the following interesting puzzle.

As we have seen above, for any linear operad $O$, the free $O$-algebra with one generator becomes a pre-Lie algebra. But the operad for pre-Lie algebra is an operad of this type. So, the free pre-Lie algebra on one generator becomes a pre-Lie algebra in this way. But of course it already \emph{is} a pre-Lie algebra! Do these pre-Lie structures agree?

The answer is \emph{no.} For an explanation, see page 7 here:

\begin{itemize}%
\item Dominique Manchon, A short survey on pre-Lie algebras. (\href{http://math.univ-bpclermont.fr/~manchon/biblio/ESI-prelie2009.pdf}{pdf})

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The best overall introduction to pre-Lie algebras seems to be that by Dominique Manchon, cited above. For two more introductions, try the following:

\begin{itemize}%
\item John Baez, This Week's Finds in Mathematical Physics, Week 299. (\href{http://math.ucr.edu/home/baez/week299.html}{web}) (\href{http://golem.ph.utexas.edu/category/2010/06/this_weeks_finds_in_mathematic_60.html}{blog})


\item Fr\'e{}d\'e{}ric Chapoton, Operadic point of view on the Hopf algebra of rooted trees. (\href{http://www-math.unice.fr/~patras/CargeseConference/ACQFT09_FredericCHAPOTON.pdf}{pdf})



\end{itemize}


\end{document}