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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lie infinity-algebroid representation} \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{representations}{Representations}\dotfill \pageref*{representations} \linebreak \noindent\hyperlink{dgcategory_of_representations}{dg-Category of representations}\dotfill \pageref*{dgcategory_of_representations} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{basic}{Basic}\dotfill \pageref*{basic} \linebreak \noindent\hyperlink{ActionOfHolomorphicTangentAlgebroid}{Action of holomorphic tangent Lie algebroid on chain complexes of complex vector bundles}\dotfill \pageref*{ActionOfHolomorphicTangentAlgebroid} \linebreak \noindent\hyperlink{Extensions}{Extensions of $L_\infty$-algebras}\dotfill \pageref*{Extensions} \linebreak \noindent\hyperlink{relations_to_comodules_over_the_ce_coalgebra}{Relations to (co-)modules over the CE (co-)algebra}\dotfill \pageref*{relations_to_comodules_over_the_ce_coalgebra} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} This entry discusses [[∞-actions]]/[[∞-representations]] of [[Lie-infinity algebroids]] (generalizing [[Lie algebra actions]]), hence the [[infinitesimal object|infinitesimal]] version of [[∞-actions]] of [[smooth ∞-groupoids]]. Recall that an $L_\infty$-[[Lie infinity-algebroid|algebroid]] is both a [[horizontal categorification]] as well as a [[vertical categorification]] of a Lie algebra: it is to Lie algebras as [[Lie ∞-groupoids]] are to Lie groups. Accordingly, the notion of \emph{representation of a Lie-$\infty$-algebroid} is a horizontal and vertical categorification of the ordinary notion of representation of a Lie algebra, which in turn is the linearization of the notion of representation of a Lie group. In view of this notice that there are essentially two fundamental ways to express the notion of [[representation]] of a [[group]] or [[∞-groupoid]] $Gr$: \begin{enumerate}% \item as a morphism out of $Gr$: the [[action]]; \item as a [[fibration sequence|fibration sequence]] over $Gr$: the [[action groupoid]]. \end{enumerate} While essentially equivalent, it is noteworthy that the first definition naturally takes place in the context of not-necessarily smooth ($\infty$-)categories, while the second one usually remains within the context of smooth ($\infty$)-groupoids: namely for $G$ a Lie group, for definiteness and for simplicity, with corresponding one-object [[Lie groupoid]] $\mathbf{B} G$ -- the [[delooping]] of the [[group]] $G$ --, a linear representation in terms of an action morphisms is a [[functor]] \begin{displaymath} \rho : \mathbf{B} G \to Vect \end{displaymath} from $\mathbf{B} G$ to the category of vector spaces. In fact, there is a canonical equivalence of the [[functor category]] $[\mathbf{B}G, Vect]$ with the category $Rep(G)$ of linear representations of $G$ \begin{displaymath} [\mathbf{B}G, Vect] \simeq Rep(G) \,. \end{displaymath} Every such functor $\rho$ induces a [[fibration sequence]] $V//G \to \mathbf{B}G$ over $\mathbf{B}G$, obtained as the [[pullback]] of the [[generalized universal bundle]] $Vect_* \to Vect$ along $\rho$ \begin{displaymath} \itexarray{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,. \end{displaymath} Here $V//G$ is the [[action groupoid]] of the [[action]] of $\rho$ on the representation vector space $V := \rho(\bullet)$, where $\bullet$ is the single object of $\mathbf{B}G$. This vector space, regarded as a [[discrete category]] on its underlying set, is the fiber of this fibration, so that the action gives rise to the [[fibration sequence|fiber sequence]] \begin{displaymath} V \hookrightarrow V//G \to \mathbf{B}G \,. \end{displaymath} As described at [[generalized universal bundle]], this may be thought of as (the groupoid incarnation of) the vector bundle which is associated via $\rho$ to the universal $G$-bundle $\mathbf{E}G \to \mathbf{B}G$, which itself is the [[action groupoid]] of the \emph{[[fundamental representation]]} of $G$ on itself, \begin{displaymath} \itexarray{ G &\hookrightarrow& \mathbf{E}G &\to& \mathbf{B}G \\ = && = && = \\ G &\hookrightarrow& G//G &\to& \mathbf{B}G } \,. \end{displaymath} From this perspective a representation of a group $G$ is nothing but a $G$-equivariant vector bundle over the point, or equivalently a vector bundle on the [[orbifold]] $\bullet//G$. So from this perspective the notion ``[[representation]]'' is not a primitive notion, but just a particular perspective on [[fibration sequences]]. The definition of Lie-$\infty$ algebroid representation below is in this [[fibration sequence]]/[[fibration]]-theoretic/[[action groupoid]] spirit. The expected alternative definition in terms of action morphisms has been considered (and is well known) apparently only for special cases. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{representations}{}\subsubsection*{{Representations}}\label{representations} Recall that we take, by definition, [[Lie ∞-algebroids]] to be [[duality|dual]] to non-negatively-graded, graded-commutative [[differential graded algebra|differential algebras]], which are free as graded-commutative algebras (qDGCAs): we write $CE_A(g)$ for the qDGCA whose underlying graded-commutative algebra is the free (over the algebra $A$) graded commutative algebra $\wedge^\bullet g^*$ for $g$ a non-postively graded cochain complex of $A$-modules and $g^*$ its degree-wise dual over $A$, to remind us that this is to be thought of as the Chevalley-Eilenberg algebra of the [[Lie ∞-algebroid]] $g$ whose space of objects is characterized dually by the algebra $A$. \begin{defn} \label{}\hypertarget{}{} A \textbf{representation} $\rho$ of a Lie $\infty$-algebroid $(g, A)$ on a co-chain complex $V$ of $A$-modules is a co[[fibration sequence]] \begin{displaymath} \wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g) \end{displaymath} in DGCAs, i.e. a homotopy pushout \begin{displaymath} \itexarray{ \wedge^\bullet V &\leftarrow& CE_\rho(g) \\ \uparrow && \uparrow \\ 0 &\leftarrow & CE_A(g) } \,. \end{displaymath} \end{defn} What has been considered in the literature so far is the more restrictive version, where the pushout is taken to be strict ([[Urs Schreiber|Urs]]: at least I think that this is the right way to say it): A \textbf{proper representation} $\rho$ is a strict cofiber sequence of morphisms of DGCAs \begin{displaymath} \wedge^\bullet V \leftarrow CE_\rho(g,V) \leftarrow CE_A(g) \end{displaymath} i.e. such that \begin{itemize}% \item $CE_\rho(g,V) = CE_A(g) \otimes \wedge^\bullet V$ as GCAs \item $\wedge^\bullet V \leftarrow CE_\rho(g,V)$ is the obvious surjection; \item $CE_\rho(g,V) \leftarrow CE_A(g)$ is the obvious injection; \item the composite of both is the 0-map. \end{itemize} It follows that the differential $d_\rho$ on $CE_\rho(g,V)$ is given by a \textbf{twisting map} $\rho^* : V \to (\wedge^\bullet V) \wedge (g^*) \wedge (\wedge^\bullet g^*)$ as \begin{itemize}% \item $d_\rho|_{g^*} = d_g$ \item $d_\rho|_{V} = d_V + \rho^*$ \end{itemize} which may be thought of as the dual of the representation morphism (see the examples below). \hypertarget{dgcategory_of_representations}{}\subsubsection*{{dg-Category of representations}}\label{dgcategory_of_representations} In (\hyperlink{Block05}{Block 05}) the [[dg-category]] $Rep(g,A)$ of proper representations of a Lie-$\infty$-algebroid $(g,A)$ in the above sense -- called \emph{dg-algebra modules} there -- is defined. \begin{defn} \label{}\hypertarget{}{} Given two objects $CE_\rho(g,V)$ and $CE_{\rho'}(g,V')$ in $Rep(g,A)$, the cochain complex $Hom( CE_\rho(g,V), CE_{\rho'}(g,V') )$ consist in degree $k$ of morphisms of degree $k$ \begin{displaymath} \phi : V \otimes \wedge^\bullet g \to V' \otimes \wedge^\bullet g^* \end{displaymath} satisfying $\phi(v t) = (-1)^{k |a|} \phi(v) t$ and the differential $d_{Hom}$ is the usual differential on hom-complexes $d \phi = d_{\rho'} \circ \phi - (-1)^{|\phi|} \phi \circ d_\rho$. For a fixed Lie $\infty$-algebroid $(g,A)$, the category \begin{displaymath} Rep(g,A) \end{displaymath} with Lie representations of $(g,A)$ as objects and chain comoplexes as above as hom-objects is a [[dg-category]]. \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{basic}{}\subsubsection*{{Basic}}\label{basic} \begin{example} \label{}\hypertarget{}{} For an n ordinary [[Lie algebra representation]] $\rho$ on a vector space $V$ consider the [[Chevalley-Eilenberg algebra]] $CE_\rho(g,V)$ that computes the [[Lie algebra cohomology]] of $\mathfrak{g}$ with [[coefficients]] in $V$. This exhibits the action in the above sense. \end{example} \begin{example} \label{ActionOfTangentLieAlgebroid}\hypertarget{ActionOfTangentLieAlgebroid}{} A [[flat connections]] on a [[vector bundle]] exhibits a representation of the [[tangent Lie algebroid]] of the base manifold. \end{example} A holomorphic variant of this is \hyperlink{ActionOfHolomorphicTangentAlgebroid}{below}. \begin{itemize}% \item (\ldots{}) [[adjoint representation]] of $L_\infty$-[[L-∞-algebras|algebras]] \end{itemize} \hypertarget{ActionOfHolomorphicTangentAlgebroid}{}\subsubsection*{{Action of holomorphic tangent Lie algebroid on chain complexes of complex vector bundles}}\label{ActionOfHolomorphicTangentAlgebroid} The following variant of example \ref{ActionOfTangentLieAlgebroid} is a [[homotopy theory|homotopy-theoretic]]-refinement of the classical [[Koszul-Malgrange theorem]]. \begin{theorem} \label{}\hypertarget{}{} For $X$ a [[smooth manifold|smooth]] [[complex manifold]] and $(g,A) = T_{hol} X$ the holomorphic [[tangent Lie algebroid]] of $X$ (so that $CE_A(g) = \Omega^\bullet_{hol}(X) = \Omega^{\bullet,0}(X)$ the holomorphic part of the [[Dolbeault complex]] of $X$), and for $Rep(T_{hol} X)$ taken to have as objects complexes of \emph{finitely generated} and \emph{projective} $C^\infty(X)$-modules (i.e. complexes of smooth [[vector bundles]]) the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho Rep(T_{hol} X)$ of the [[dg-category]] $Rep(T_{hol} X)$ is [[equivalence of categories|equivalent]] to the \emph{bounded [[derived category]] of [[chain complexes]] of [[abelian sheaves]] with [[coherent cohomology]]} on $X$ (see at \emph{[[coherent sheaf]]}). \end{theorem} This is (\hyperlink{Block05}{Block 05, theorem 2.22} (in the counting of version 1 on the arXiv!)). The objects of $Rep(T_{hol} X)$ are literally complexes of smooth vector bundles that are equipped with ``half a flat connection'', namely with a flat covariant derivative only along holomorphic tangent vectors. It is an old result that [[holomorphic vector bundles]] (see there) are equivalent to such smooth vector bundles with ``half a flat connection''. This is what the theorem is based on. \hypertarget{Extensions}{}\subsubsection*{{Extensions of $L_\infty$-algebras}}\label{Extensions} \begin{example} \label{}\hypertarget{}{} For $\mathfrak{g}$ any [[L-∞ algebra]], and $\mathfrak{a}$ any other,then an \href{infinity-Lie+algebra+cohomology#Extensions}{L-∞ extension} (see there) $\hat {\mathfrak{g}}$ of $\mathfrak{g}$ by $\mathfrak{a}$ is a [[homotopy fiber sequence]] \begin{displaymath} \mathfrak{a} \to \hat {\mathfrak{g}} \to \mathfrak{g} \end{displaymath} of [[L-∞ algebras]] (see at \emph{[[model structure for L-∞ algebras]]}). Regarding this as sequence of [[L-∞ algebroids]] over the point \begin{displaymath} \mathbf{B}\mathfrak{a} \to \mathbf{B}\hat {\mathfrak{g}} \to \mathbf{B}\mathfrak{g} \end{displaymath} and then passing to [[Chevalley-Eilenberg algebras]], this exhibits an action/representation of the $L_\infty$-algebra $\mathfrak{g}$ on the $L_\infty$-algebroid $\mathbf{B}\mathfrak{a}$. \end{example} For instance the [[string Lie 2-algebra]] is the $\mathbf{B} \mathbb{R}$-extension of a semisimple [[Lie algebra]] $\mathfrak{g}$ with bilinear [[invariant polynomial]] $\langle -,-\rangle$ corresponding to the 3-cocycle $\langle -,[-,-]\rangle \in CE(\mathfrak{g})$, hence exhibits an action/representation of $\mathfrak{g}$ on $\mathbf{B}\mathbb{R}$. This is the infinitesimal version of the [[∞-action]] of a simply connected compact simple [[Lie group]] $G$ on the [[circle 2-group]] $\mathbf{B}U(1)$ which exhibits the [[String 2-group]] extension. Analogous statements in various degrees hold for the $L_\infty$-algebra [[Fivebrane 6-group]] \begin{displaymath} \mathbf{B}^6 \mathbb{R} \to \mathfrak{fivebrane}\to \mathfrak{string} \end{displaymath} exhibiting an $\infty$-action of the [[string Lie 2-algebra]] on $\mathbf{B}^7 \mathbb{R}$, and analogously for the [[supergravity Lie 3-algebra]], the [[supergravity Lie 6-algebra]] and for all the other extensions in [[schreiber:The brane bouquet]]. \hypertarget{relations_to_comodules_over_the_ce_coalgebra}{}\subsection*{{Relations to (co-)modules over the CE (co-)algebra}}\label{relations_to_comodules_over_the_ce_coalgebra} Under identifying [[L-infinity algebras]] with graded (co-)commutative [[dg-coalgebras]]/[[dg-algebras]] (their [[Chevalley-Eilenberg algebras]]) then their representations correspond to dg-(co-)modules whose underlying graded (co-)algebras are (co-)free. See at \begin{itemize}% \item [[model structure on dg-comodules]] \item [[model structure on dg-modules]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} The definition of representation of $L_\infty$-algebras is discussed in section 5 of \begin{itemize}% \item [[Tom Lada]], [[Martin Markl]], \emph{Strongly homotopy Lie algebras} (\href{http://arxiv.org/abs/hep-th/9406095}{arXiv:9406095}) \end{itemize} The general definition of representation of $\infty$-Lie algebroids (of [[finite type]]) as above appears as def. 4.9 in \begin{itemize}% \item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Differential twisted string- and fivebrane structures} (\href{http://arxiv.org/abs/0910.4001v1}{arXiv:0910.4001 version 1}) \end{itemize} (this discussion is not in the published version \href{http://arxiv.org/abs/0910.4001v2}{arXiv:0910.4001v2}, for size reasons) modeled after the geneal abstract definition of [[∞-actions]] in \begin{itemize}% \item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]}, Journal of Homotopy and Related Structures, 2014 (\href{http://arxiv.org/abs/1207.0248}{arXiv:1207.0248}) \end{itemize} The definition of the dg-category of representation of a tangent Lie algebroid and its equivalence in special cases to derived categories of complexes of coherent sheaves is in \begin{itemize}% \item [[Jonathan Block]], \emph{Duality and equivalence of module categories in noncommutative geometry I} (\href{http://arxiv.org/abs/math/0509284}{arXiv:0509284}) \end{itemize} Application of this to the description of [[B-branes]] is in \begin{itemize}% \item [[Aaron Bergman]], \emph{Topological D-branes from Descent} (\href{http://arxiv.org/abs/0808.0168}{arXiv:0808.0168}) \end{itemize} For the case of Lie 1-algebroids essentially the same definition appears also in \begin{itemize}% \item [[Camilo Arias Abad]], [[Marius Crainic]], \emph{Representations up to homotopy of Lie algebroids} (\href{http://arxiv.org/abs/0901.0319}{arXiv}) \end{itemize} The [[Lie integration]] of representations of Lie 1-algebroids $\mathfrak{a} \to end(V)$ to morphisms of [[∞-categories]] $A \to Ch_\bullet^\circ$ is discussed in \begin{itemize}% \item [[Camilo Arias Abad]], [[Florian Schätz]], \emph{The $A_\infty$ de Rham theorem and integration of representations up to homotopy} (\href{http://arxiv.org/abs/1011.4693}{arXiv:1011.4693}) \end{itemize} [[!redirects representation of an ∞-Lie algebroid]] [[!redirects representations of ∞-Lie algebroids]] [[!redirects L-∞ algebroid representation]] [[!redirects L-∞ algebroid representations]] [[!redirects L-infinity algebroid representation]] [[!redirects L-infinity algebroid representations]] [[!redirects Lie infinity-algebroid representations]] [[!redirects Lie ∞-algebroid representation]] [[!redirects Lie-∞ algebroid representation]] [[!redirects Lie-∞ algebroid representations]] [[!redirects Lie ∞-algebroid representations]] [[!redirects representation up to homotopy]] [[!redirects representations up to homotopy]] [[!redirects ∞-Lie algebroid representation]] [[!redirects ∞-Lie algebroid representations]] [[!redirects L-∞ algebra action]] [[!redirects L-∞ algebra actions]] [[!redirects L-infinity algebra action]] [[!redirects L-infinity algebra actions]] \end{document}