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\begin{document}

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\section*{Lie-Rinehart pair}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{Lie--Rinehart pair} is an algebraic encoding of the notion of \emph{[[Lie algebroid]]}. It is the pair consisting of the [[associative algebra]] of [[functions]] on the base space of the Lie algebroid and of the [[Lie algebra]] of its global [[sections]]. The [[anchor map]] of the Lie algebroid is encoded in the [[action]] of the Lie algebra on the associative algebra by [[derivations]] and the local structure is encoded in the Lie algebra being a [[module]] over the associative algebra.

Since in this formulation the base manifold of the Lie algebroid is entirely described [[Isbell duality|dually]] in terms of its [[algebra of functions]], and since the definition does not refer to this being a \emph{commutative} algebra, the notion of Lie-Rinehart pair in fact generalizes the notion of Lie algebroid from ordinary [[differential geometry]] to [[noncommutative geometry]].

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

A \emph{Lie--Rinehart-pair} $(A,\mathfrak{g})$ is a pair consisting of

\begin{enumerate}%
\item an [[associative algebra]] $A$


\item a [[Lie algebra]] $\mathfrak{g}$



\end{enumerate}
such that

\begin{enumerate}%
\item $A$ is a $\mathfrak{g}$-[[module]]


\item $\mathfrak{g}$ is an $A$-[[module]]



\end{enumerate}
with both module structures being compatible in the obvious way:

\begin{enumerate}%
\item $\mathfrak{g}$ acts as [[derivations]] of $A$: that is, we have a Lie algebra [[homomorphism]] $\mathfrak{g} \to Der(A)$.


\item $A$ acts as linear transformations of $\mathfrak{g}$ in a way obeying the [[Leibniz rule]]: that is, we have an associative algebra homomorphism $A \to End(\mathfrak{g})$, where $End(\mathfrak{g})$ is the algebra of all linear transformations of $\mathfrak{g}$, such that

\begin{displaymath}
[v, a w] = v(a) w + a [v,w].
\end{displaymath}


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

In the case that $A = C^\infty(X)$ is the algebra of smooth functions on a smooth [[manifold]] $X$, Lie--Rinehart pairs $(C^\infty(X), \mathfrak{g})$ are naturally identified with [[Lie algebroid]]s over $X$: given the [[Lie algebroid]] in its incarnation as a [[vector bundle]] morphism

\begin{displaymath}
\itexarray{
     E &&\stackrel{\rho}{\to}&& T X
     \\
     & \searrow && \swarrow
     \\
     && X 
  }
\end{displaymath}
equipped with a bracket

\begin{displaymath}
[-,-] : \Gamma(E) \otimes\Gamma(E) \to \Gamma(E)
\end{displaymath}
we obtain a Lie--Rinehart pair by setting

\begin{itemize}%
\item $\mathfrak{g} = \Gamma(E)$ is the [[Lie algebra]] of [[sections]] of $E$ using the above bracket


\item the action of $A$ on $\mathfrak{g}$ is the obvious multiplication of sections of vector bundles over $X$ by functions on $X$


\item the action of $\mathfrak{g}$ on $C^\infty(X)$ is given by first applying the anchor map $\rho$ and then using the canonical action of vector fields on functions.



\end{itemize}
So for all the examples listed at [[Lie algebroid]] we obtain an example for Lie--Rinehart pairs.

In particular

\begin{itemize}%
\item the Lie--Rinehart pair coresponding to the [[tangent Lie algebroid]] of a [[manifold]] $X$ is $(C^\infty(X), \Gamma(T X))$ with the obvious action on each other.


\item the Lie--Rinehart pair corresponding to an ordinary [[Lie algebra]] $\mathfrak{g}$ is $(\mathbb{R}, \mathfrak{g})$ with $\mathfrak{g}$ acting trivially on $\mathbb{R}$.


\item the Lie--Rinehart pair corresponding to a [[Poisson Lie algebroid]] on a [[Poisson manifold]] $X$ is $(C^\infty(X), MultVect(X))$, where the [[Lie algebra]] is the the space of [[multivector fields]] on $X$ equipped with the [[Schouten bracket]].



\end{itemize}
In \href{string%20field%20theory#OpenClosedStringFieldTheory}{open-closed string field theory} one finds at least one half of the axioms of homotopy Lie-Rinehart pairs.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

A little bit is known in the literature to generalizations of the notion of Lie--Rinehart algebras that are to [[Lie ∞-algebroids]] as the latter are to [[Lie algebroids]].

In

\begin{itemize}%
\item [[Dmitry Roytenberg]], \emph{Courant--Dorfman algebras and their cohomology} (\href{http://arxiv.org/abs/0902.4862}{arXiv})

\end{itemize}
the analogous algebraic structure for [[Courant algebroid]]s is discussed. These ``2-Lie--Rinehart algebras'' are called [[Courant?Dorfman algebra]]s there.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[bisections of a Lie groupoid]]

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

The original reference is

\begin{itemize}%
\item G. Rinehart, \emph{Differential forms for general commutative algebras}, Trans. Amer. Math. Soc. \textbf{108} (1963), 195-222

\end{itemize}
A brief review in section 1 of

\begin{itemize}%
\item [[Johannes Huebschmann]], \emph{Lie--Rinehart algebras, descent, and quantization} (\href{http://arxiv.org/abs/math/0303016}{arXiv})


\item [[Mikhail Kapranov|M. Kapranov]], \emph{Free Lie algebroids and the space of paths}, Sel. Math. (N.S.) \textbf{13}, n. 2 277--319 (2007), \href{http://arxiv.org/abs/math.AG/0702584}{arXiv:math.AG/0702584}, \href{http://dx.doi.org/10.1007/s00029-007-0041-9}{doi}


\item V. Nistor, [[Alan Weinstein]], [[Ping Xu]], \emph{Pseudodifferential operators on differential groupoids}, Pacific J. Math. \textbf{189}, 117--152 (1999)



\end{itemize}
A notion of universal enveloping algebra of a Lie--Rinehart algebra is discussed in

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Janez Mrčun]], \emph{On the universal enveloping algebra of a Lie--Rinehart algebra}, Proc. Amer. Math. Soc. in press, (\href{http://www.ams.org/proc/0000-000-00/S0002-9939-10-10347-5/S0002-9939-10-10347-5.pdf}{pdf}, \href{http://arxiv.org/abs/0801.3929}{arXiv/0801.3929})

\end{itemize}
A connection with [[BV-theory]] and [[L-infinity algebra]] is made in

\begin{itemize}%
\item [[Lars Kjeseth]], \emph{Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs}, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (\href{https://projecteuclid.org/euclid.hha/1140370269}{Euclid})


\item [[Johannes Huebschmann]], \emph{Lie--Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras} (\href{http://aif.cedram.org/item?id=AIF_1998__48_2_425_0}{journal})



\end{itemize}
[[!redirects Lie-Rinehart pairs]]

[[!redirects Lie-Rinehart algebra]] [[!redirects Lie?Rinehart pair]] [[!redirects Lie?Rinehart algebra]] [[!redirects Lie--Rinehart pair]] [[!redirects Lie--Rinehart algebra]]

[[!redirects Lie-Rinehart algebras]] [[!redirects Lie?Rinehart pairs]] [[!redirects Lie?Rinehart algebras]] [[!redirects Lie--Rinehart pairs]] [[!redirects Lie--Rinehart algebras]]

[[!redirects homotopy Lie-Rinehart pair]] [[!redirects homotopy Lie-Rinehart pairs]]



\end{document}