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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*{Poisson bracket Lie n-algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{the_extension_theorem}{The extension theorem}\dotfill \pageref*{the_extension_theorem} \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 [[Lie n-algebra]] that generalizes the [[Poisson bracket]] from [[symplectic geometry]] to [[n-plectic geometry]]: the \emph{Poisson bracket $L_\infty$-algebra of local observables} in [[higher prequantum geometry]]. More discussion is \href{n-plectic+geometry#PoissonLInfinityAlgebras}{here} at \emph{[[n-plectic geometry]]}. Applied to the symplectic current (in the sense of [[covariant phase space]] theory, [[de Donder-Weyl field theory]]) this is the higher [[current algebra]] (see there) of [[conserved currents]] of a [[prequantum field theory]]. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} Throughout, Let $X$ be a [[smooth manifold]], let $n \geq 1$ a natural number and $\omega \in \Omega^{n+1}_{cl}(X)$ a closed [[differential n-form|differential (n+1)-form]] on $X$. The pair $(X,\omega)$ we may regard as a [[pre-n-plectic manifold]]. We define two [[L-∞ algebras]] defined from this data and discuss their [[equivalence]]. Either of the two or any further one equivalent to the two is the \emph{Poisson bracket Lie $n$-albebra} of $(X,\omega)$. The first definition is due to (\hyperlink{Rogers10}{Rogers 10}), the second due to (\hyperlink{FRS13b}{FRS 13b}). Here in notation we follow (\hyperlink{FRS13b}{FRS 13b}). \begin{defn} \label{HamiltonianFormsAndVectorFields}\hypertarget{HamiltonianFormsAndVectorFields}{} Write \begin{displaymath} Ham^{n-1}(X) \subset Vect(X) \oplus \Omega^{n-1}(X) \end{displaymath} for the subspace of the [[direct sum]] of [[vector fields]] $v$ on $X$ and [[differential n-form|differential (n-1)-forms]] $J$ on $X$ satisfying \begin{displaymath} \iota_v \omega + \mathbf{d} J = 0 \,. \end{displaymath} We call these the \emph{pairs of [[Hamiltonian forms]] with their [[Hamiltonian vector fields]]}. \end{defn} (\hyperlink{FRS13b}{FRS 13b, def. 2.1.3}) \begin{defn} \label{PoissonBracketLienAlgebra}\hypertarget{PoissonBracketLienAlgebra}{} The [[L-∞ algebra]] $L_\infty(X,\omega)$ has as underlying [[chain complex]] the truncated and modified [[de Rham complex]] \begin{displaymath} \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^{n-2}(X) \stackrel{(0,\mathbf{d})}{\longrightarrow} Ham^{n-1}(X) \end{displaymath} with the Hamiltonian pairs, def. \ref{HamiltonianFormsAndVectorFields}, in degree 0 and with the 0-forms ([[smooth functions]]) in degree $n-1$, and its non-vanishing $L_\infty$-brackets are as follows: \begin{itemize}% \item $l_1(J) = \mathbf{d}J$ \item $l_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) = - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_1 \wedge \cdots \wedge v_k}\omega$. \end{itemize} \end{defn} (\hyperlink{FRS13b}{FRS 13b, prop. 3.1.2}) \begin{defn} \label{PoissondgAlgebra}\hypertarget{PoissondgAlgebra}{} Let $\overline{A}$ be any [[Cech cohomology|Cech]]-[[Deligne cohomology|Deligne]]-[[cocycle]] relative to an [[open cover]] $\mathcal{U}$ of $X$, which gives a [[prequantum n-bundle]] for $\omega$. The [[L-∞ algebra]] $dgLie_{Qu}(X,\overline{A})$ is the [[dg-Lie algebra]] (regarded as an $L_\infty$-algebra) whose underlying [[chain complex]] is $dgLie_{Qu}(X,\overline{A})^0 = \{v+ \overline{\theta} \in Vect(X)\oplus Tot^{n-1}(\mathcal{U}, \Omega^\bullet) \;\vert\; \mathcal{L}_v \overline{A} = \mathbf{d}_{Tot}\overline{\theta}\}$; $dgLie_{Qu}(X,\overline{A})^{i \gt 0} = Tot^{n-1-i}(\mathcal{U},\Omega^\bullet)$ with [[differential]] given by $\mathbf{d}_{Tot}$ (where $Tot$ refers to [[total complex]] of the Cech-de Rham [[double complex]]). The non-vanishing dg-Lie bracket on this complex are defined to be \begin{itemize}% \item $[v_1 + \overline{\theta}_1, v_2 + \overline{\theta}_2] = [v_1, v_2] + \mathcal{L}_{v_1}\overline{\theta}_2 - \mathcal{L}_{v_2}\overline{\theta}_1$; \item $[v+ \overline{\theta}, \overline{\eta}] = - [\eta, v + \overline{\theta}] = \mathcal{L}_v \overline{\eta}$. \end{itemize} \end{defn} (\hyperlink{FRS13b}{FRS 13b, def./prop. 4.2.1}) \begin{prop} \label{ComparisonTheorem}\hypertarget{ComparisonTheorem}{} There is an [[equivalence]] in the [[model structure for L-∞ algebras|homotopy theory of L-∞ algebras]] \begin{displaymath} f \colon L_\infty(X,\omega) \stackrel{\simeq}{\longrightarrow} dgLie_{Qu}(X,\overline{A}) \end{displaymath} between the $L_\infty$-algebras of def. \ref{PoissonBracketLienAlgebra} and def. \ref{PoissondgAlgebra} (in particular def. \ref{PoissondgAlgebra} does not depend on the choice of $\overline{A}$) whose underlying [[chain map]] satisfies \begin{itemize}% \item $f(v + J) = v - J|_{\mathcal{U}} + \sum_{i = 0}^n (-1)^i \iota_v A^{n-i}$. \end{itemize} \end{prop} (\hyperlink{FRS13b}{FRS 13b, theorem 4.2.2}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{the_extension_theorem}{}\subsubsection*{{The extension theorem}}\label{the_extension_theorem} \begin{defn} \label{ExtensionTheorem}\hypertarget{ExtensionTheorem}{} Given a [[pre n-plectic manifold]] $(X,\omega_{n+1})$, then the Poisson bracket Lie $n$-algebra $\mathfrak{Pois}(X,\omega)$ from \hyperlink{Definition}{above} is an [[L-infinity algebra cohomology|extension]] of the [[Lie algebra]] of [[Hamiltonian vector fields]] $Vect_{Ham}(X)$, def. \ref{HamiltonianFormsAndVectorFields} by the [[cocycle]] [[infinity-groupoid]] $\mathbf{H}(X,\flat \mathbf{B}^{n-1} \mathbb{R})$ for [[ordinary cohomology]] with [[real number]] [[coefficients]] in that there is a [[homotopy fiber sequence]] in the [[homotopy theory of L-infinity algebras]] of the form \begin{displaymath} \itexarray{ \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) &\longrightarrow& \mathfrak{Pois}(X,\omega) \\ && \downarrow \\ && Vect_{Ham}(X,\omega) &\stackrel{\omega[\bullet]}{\longrightarrow}& \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) } \,, \end{displaymath} where the [[cocycle]] $\omega[\bullet]$, when realized on the model of def. \ref{PoissonBracketLienAlgebra}, is degreewise given by by contraction with $\omega$. \end{defn} This is \hyperlink{FRS13b}{FRS13b, theorem 3.3.1}. As a corollary this means that the [[0-truncation]] $\tau_0 \mathfrak{Pois}(X,\omega)$ is a [[Lie algebra extension]] by [[de Rham cohomology]], fitting into a [[short exact sequence]] of [[Lie algebras]] \begin{displaymath} 0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,. \end{displaymath} \begin{remark} \label{}\hypertarget{}{} These kinds of extensions are known traditionally form [[current algebras]]. \end{remark} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[higher Poisson structure]] \begin{itemize}% \item [[Poisson n-algebra]] \item [[Nambu bracket]] \end{itemize} \item [[current algebra]] \item [[geometry of physics -- prequantum geometry]] \end{itemize} [[!include slice automorphism groups in higher prequantum geometry - table]] [[!include geometric quantization extensions - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The Poisson bracket $L_\infty$-algebra $L_\infty(X,\omega)$ was introduced in \begin{itemize}% \item [[Chris Rogers]], \emph{$L_\infty$ algebras from multisymplectic geometry}, Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (\href{http://arxiv.org/abs/1005.2230}{arXiv:1005.2230}, \href{http://link.springer.com/article/10.1007%2Fs11005-011-0493-x}{journal}). \item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068}) \end{itemize} Discussion in the broader context of [[higher differential geometry]] and [[higher prequantum geometry]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory|Higher $U(1)$-gerbe connections in geometric prequantization]]}, Rev. Math. Phys., Vol. 28, Issue 06, 1650012 (2016) (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236}) \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]}, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 -- 142 (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292}) \item Nestor Leon Delgado, \emph{Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables} (\href{https://arxiv.org/abs/1805.10317}{arXiv:1805.10317}) \end{itemize} See also \begin{itemize}% \item [[Patricia Ritter]], [[Christian Sämann]], \emph{Automorphisms of Strong Homotopy Lie Algebras of Local Observables} (\href{http://arxiv.org/abs/1507.00972}{arXiv:1507.00972}) \end{itemize} [[!redirects Poisson bracket Lie n-algebras]] [[!redirects Poisson-bracket Lie n-algebra]] [[!redirects Poisson-bracket Lie n-algebras]] [[!redirects Poisson Lie n-algebra]] [[!redirects Poisson Lie n-algebras]] [[!redirects Poisson L-∞ algebra]] [[!redirects Poisson L-∞ algebras]] [[!redirects Poisson bracket L-∞ algebra]] [[!redirects Poisson brakcet L-∞ algebras]] [[!redirects Poisson L-infinity algebra]] [[!redirects Poisson L-infinity algebras]] [[!redirects Poisson-bracket Lie n-algebra of local observables]] [[!redirects Poisson-bracket Lie n-algebras of local observables]] [[!redirects L-∞ algebra of local observables]] [[!redirects L-∞ algebras of local observables]] [[!redirects Poisson bracket Lie (p+1)-algebra]] [[!redirects Poisson bracket Lie (p+1)-algebras]] \end{document}