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\section*{Atiyah Lie algebroid}

\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{relation_to_connections}{Relation to connections}\dotfill \pageref*{relation_to_connections} \linebreak
\noindent\hyperlink{atiyah_class}{Atiyah class}\dotfill \pageref*{atiyah_class} \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 \textbf{Atiyah Lie algebroid} associated to a $G$-[[principal bundle]] $P$ over $X$ is a [[Lie algebroid]] structure on the [[vector bundle]] $T P/ G$, the [[quotient]] of the [[tangent bundle]] of the total space $P$ by the canonical induced $G$-[[action]].

The [[Lie groupoid]] that the Atiyah Lie algebroid [[Lie integration|integrates to]] is the \emph{[[Atiyah Lie groupoid]]}. See there for more background and discussion.

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

Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and let $P \to X$ be a $G$-[[principal bundle]]:

the \textbf{Atiyah Lie algebroid sequence} of $P$ is a sequence of [[Lie algebroid]]s

\begin{displaymath}
ad(P) \to at(P) \to T X
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $ad(P) = P \times_G \mathfrak{g}$ is the [[adjoint bundle]] of Lie algebras, associated via the [[adjoint action]] of $G$ on its Lie algebra;


\item $at(P) := (T P)/G$ is the \textbf{Atiyah Lie algebroid}


\item $T X$ is the [[tangent Lie algebroid]] of $X$.



\end{itemize}
The [[Lie bracket]] on the sections of $at(P)$ is that inherited from the tangent Lie algebroid of $P$.

\hypertarget{relation_to_connections}{}\subsection*{{Relation to connections}}\label{relation_to_connections}

A splitting $\nabla_{flat} : T X \to at(P)$ of the Atiyah Lie algebroid sequence in the category of [[Lie algebroid]]s is precisely a flat [[connection on a bundle|connection on]] $P$.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of [[vector bundle]]s. In that case one finds the curvature of the connection precisely as the [[obstruction]] to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_\infty$-[[L-infinity-algebroid|algebroids]], in terms of an [[horizontal categorification]] of [[nonabelian Lie algebra cohomology]]:

\hypertarget{atiyah_class}{}\subsection*{{Atiyah class}}\label{atiyah_class}

The $Ext^1$-cohomology class corresponding to the Atiyah exact sequence (usually in a version for vector bundles/coherent sheaves) is the \textbf{Atiyah class}.

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

\begin{itemize}%
\item [[Courant Lie 2-algebroid]]

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

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Complex analytic connections in fibre bundles}, Trans. Amer. Math. Soc. 85 (1957), 181--207, \href{http://dx.doi.org/10.2307/1992969}{doi},\href{http://www.ams.org/mathscinet-getitem?mr=0086359}{MR0086359}
\item Pietro Tortella, \emph{Representations of Atiyah algebroids and logarithmic connections}, \href{http://arxiv.org/abs/1505.04763}{arxiv/1505.04763}

\end{itemize}
A discussion with an emphasis on the relation to [[connection on a bundle|connections]] and [[Lie 2-algebra]]s is on the first pages of

\begin{itemize}%
\item [[Danny Stevenson]], Lie 2-algebras and the geometry of gerbes, Unni Namboodiri Lectures 2006 \href{http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf}{slides}

\end{itemize}
For Atiyah classes see

\begin{itemize}%
\item L. Illusie, \emph{Complexe cotangent et d\'e{}formations} (vol. 1) IV.2.3
\item \href{http://mathoverflow.net/questions/56405/atiyah-class-for-non-locally-free-sheaf}{MO:atiyah-class-for-non-locally-free-sheaf}
\item [[M. Kapranov]], \emph{Rozansky--Witten invariants via Atiyah classes}, Compositio Math. 115 (1999), 71--113.
\item U. Bruzzo, I. Mencattini, V. Rubtsov, P. Tortella, \emph{Nonabelian Lie algebroid extensions}, arXiv:1305.2377.
\item Zhuo Chen, Mathieu Sti\'e{}non, Ping Xu, \emph{From Atiyah classes to homotopy Leibniz algebras}, \href{http://arxiv.org/abs/1204.1075}{arXiv/1204.1075}; \emph{A Hopf algebra associated to a Lie pair}, \href{http://arxiv.org/abs/1409.6803}{arxiv/1409.6803}
\item R. A. Mehta, M. Sti\'e{}non, P. Xu, \emph{The Atiyah class of a dg-vector bundle}, \href{http://arxiv.org/abs/1502.03119}{arxiv/1502.03119}
\item [[Nikita Markarian]], \emph{The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem}, J. Lond. Math. Soc. (2) 79 (2009), no. 1, 129--143
\item F. Bottacin, \emph{Atiyah classes for Lie algebroids}, \href{http://www.math.unipd.it/~bottacin/papers/liealgebroids.pdf}{pdf}
\item Ajay C. Ramadoss, The big Chern classes and the Chern character, Internat. J. Math. 19 (2008), no. 6, 699--746.

\end{itemize}
[[!redirects Atiyah Lie algebroids]] [[!redirects Atiyah algebroid]] [[!redirects Atiyah algebroids]] [[!redirects Atiyah class]] [[!redirects Atiyah sequence]]



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