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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*{Heisenberg Lie algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{InSymplecticGeometry}{In symplectic geometry}\dotfill \pageref*{InSymplecticGeometry} \linebreak \noindent\hyperlink{InnplecticGeometry}{Heisenberg Lie $n$-algebras in $n$-plectic geometry}\dotfill \pageref*{InnplecticGeometry} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{RelationToPoissonAlgebra}{Relation to Poisson algebra}\dotfill \pageref*{RelationToPoissonAlgebra} \linebreak \noindent\hyperlink{integration_to_heisenberg_group}{Integration to Heisenberg group}\dotfill \pageref*{integration_to_heisenberg_group} \linebreak \noindent\hyperlink{relation_to_the_weyl_algebra}{Relation to the Weyl algebra}\dotfill \pageref*{relation_to_the_weyl_algebra} \linebreak \noindent\hyperlink{relation_to_the_heisenberg_double}{Relation to the Heisenberg double}\dotfill \pageref*{relation_to_the_heisenberg_double} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Suppose we are given a commutative unital [[ring]] $k$ and a [[module]] $V$ over $k$ equipped with a skew-symmetric [[bilinear form]] \begin{displaymath} \omega: V\otimes_k V\to k \,. \end{displaymath} (Typically, one requires $\omega$ to be non-degenerate, see \href{InSymplecticGeometry}{below}, but this is not needed for the following definition). The \textbf{Heisenberg Lie algebra} $Heis(V, \omega)$ corresponding to $(V,\omega)$ is the [[Lie algebra]] given by the $k$-module $V\oplus k$ together with the unit $k \hookrightarrow V\oplus k$, $s\mapsto (0,s) =: s 1$ and Lie $k$-algebra bracket \begin{displaymath} [(m,s),(m',s')] := (0, \omega(m,m')1) \,. \end{displaymath} \hypertarget{InSymplecticGeometry}{}\subsubsection*{{In symplectic geometry}}\label{InSymplecticGeometry} The notion of Heisenberg algebra arose in the study of [[quantization]] by tools of [[symplectic geometry]]: A special case of the above definition is that where $(V,\omega)$ a [[symplectic vector space]] (hence $k$ a [[field]] and $\omega$ non-degenerate). In this case the Heisenberg algebra is a sub-Lie algebra of the Lie algebra underlying the [[Poisson algebra]] of $(V,\omega)$. For more on this see \hyperlink{RelationToPoissonAlgebra}{below}. \hypertarget{InnplecticGeometry}{}\subsubsection*{{Heisenberg Lie $n$-algebras in $n$-plectic geometry}}\label{InnplecticGeometry} We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary [[symplectic geometry]] to a notion of \emph{Heisenberg [[Lie n-algebra]]} in [[higher geometric quantization]] of [[n-plectic geometry]]. See at \emph{[[Heisenberg Lie n-algebra]]} for more. The following definition is naturally motivated from the fact that: \begin{enumerate}% \item The ordinary Heisenberg Lie algebra is the sub-Lie algebra of the [[Poisson bracket]] Lie algebra, the one underlying the corresponding [[Poisson algebra]] (see \hyperlink{RelationToPoissonAlgebra}{below}) on the constant and linear functions. \item The generalization of [[Poisson bracket|Poisson brackets]] to [[Poisson Lie n-algebras]] in [[n-plectic geometry]] for all $n$ is established (see there). \end{enumerate} In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-algebra of the [[Poisson Lie n-algebra]] on the linear and constant differential forms. First we need the following definition, which is elementary, but nevertheless worth making explicit once. \begin{defn} \label{}\hypertarget{}{} Let $n \in \mathbb{N}$, let $(V, \omega)$ be an [[n-plectic vector space]]. The \textbf{corresponding $n$-plectic manifold} is the [[n-plectic manifold]] $(V, \mathbf{\omega})$, with $V$ now the canonical [[smooth manifold]] structure on the given vector space, and with \begin{displaymath} \mathbf{\omega} \in \Omega^{n+1}(V) \end{displaymath} the [[differential form]] obtained by left (right) translating $\omega$ along $V$. Explicitly, for all [[vector fields]] $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set \begin{displaymath} \mathbf{\omega}_x(v_1, \cdots, v_n) := \omega(v_1(x), \cdots, v_n(x)) \,. \end{displaymath} Here on the right -- and in all of the following -- we are using that every [[tangent space]] $T_x V$ of $V$ is naturally identified with $V$ itself \begin{displaymath} T_x V \stackrel{\simeq}{\to} V \,. \end{displaymath} \end{defn} \begin{defn} \label{}\hypertarget{}{} Let $n \in \mathbb{N}$, let $(V, \omega)$ be an [[n-plectic vector space]] and let $(V, \mathbf{\omega})$ be the corresponding [[n-plectic manifold]]. The \textbf{Heisenberg Lie $n$-algebra} $Heis(V,\omega)$ is the sub-[[Lie n-algebra]] of the [[Poisson Lie n-algebra]] $\mathcal{P}(V, \omega)$ on those [[differential forms]] which are either linear or constant (with respect to left/right translation on $V$). \end{defn} All one has to observe is: \begin{prop} \label{}\hypertarget{}{} This is indeed a sub-Lie $n$-algebra. \end{prop} \begin{proof} We need to check that the linear and constant forms are closed under the [[L-infinity algebra]] brackets of $\mathcal{P}(V, \omega)$. The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0. The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this. Similarly, the brackets on elements all in degree 0 is given by contraction of $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with $\mathbf{\omega}$ gives a constant form. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{RelationToPoissonAlgebra}{}\subsubsection*{{Relation to Poisson algebra}}\label{RelationToPoissonAlgebra} We discuss how the notion of Heisenberg Lie algebra relates to that of \emph{[[Poisson algebra]]}. \begin{prop} \label{}\hypertarget{}{} For $(X, \omega)$ a [[symplectic vector space]], there is a [[natural transformation|natural]] [[Lie algebra]] [[homomorphism]] \begin{displaymath} Heis(V, \omega) \hookrightarrow \mathcal{P}(V,\omega) \end{displaymath} exhibiting the Heisenberg Lie algebra as a [[subobject|sub]]-Lie algebra of that underlying the [[Poisson algebra]] $\mathcal{P}(V,\omega)$ of $V$. \end{prop} Namely, it is the sub-Lie algebra on the [[linear functions]] and the constant functions. \begin{proof} Let $(V, \omega)$ be a [[symplectic vector space]] over the [[real numbers]]. Using the canonical [[isomorphism]] $\phi : T V \simeq V \times V$ of the [[tangent bundle]] of $V$ with the [[projection]] $p_1 : V \times V \to V$, we obtain from the [[bilinear form]] $\omega$ a [[differential form|differential 2-form]] $\mathbf{\omega} \in \Omega^2(V)$ by the assignment \begin{displaymath} \mathbf{\omega}(\mathbf{v}, \mathbf{w}) := \omega(p_2 \phi(\mathbf{v}), p_2 \phi(\mathbf{w})) \end{displaymath} for all $\mathbf{v}, \mathbf{w} \in \Gamma(T V)$. This way $(V, \mathbf{\omega})$ is a [[symplectic manifold]] and thus comes with a [[Poisson algebra]]. Write $\mathcal{P}(V,\mathbf{\omega})$ for the [[Lie algebra]] underlying the [[Poisson algebra]] of $(V, \mathbf{\omega})$. Its underlying [[vector space]] is the space $C^\infty(V)$ of [[smooth functions]] $V \to \mathbb{R}$. To every element $f \in C^\infty(V)$ is associated its \emph{[[Hamiltonian vector field]]} $\mathbf{v}_f \in \Gamma(T X)$, defined (uniquely, due to the non-degeneracy of $\omega$) by the equation \begin{displaymath} d_{dR} f = \mathbf{\omega}(\mathbf{v}_f, -) \,. \end{displaymath} In terms of this, the [[Lie bracket]] of the Poisson algebra is defined to be \begin{displaymath} [f,g] := \mathbf{\omega}(\mathbf{v}_f, \mathbf{v}_g) \,. \end{displaymath} Inside all smooth functions sit the [[linear functions]] $V \to \mathbb{R}$, which form the [[dual vector space]] to $V$: \begin{displaymath} V^* \hookrightarrow C^\infty(V) \end{displaymath} \begin{displaymath} \alpha \mapsto \alpha(-) \,. \end{displaymath} By the non-degeneracy of $\omega$, for every $f \in V^*$ there is an element $v_f \in V$ such that \begin{displaymath} f = \omega(v_f,-) \in C^\infty(V) \,. \end{displaymath} Moreover, the canonical extension $\mathbf{v}_f$ of $v_f$ to a vector field on $V$ is a [[Hamiltonian vector field]] for $f$ \begin{displaymath} d_{dR} f = \mathbf{\omega}(\mathbf{v}_f,-) \,. \end{displaymath} It follows that the Lie bracket of two [[linear functions]] $f,g$ in the Poisson algebra is \begin{displaymath} \begin{aligned} [f,g] & = \mathbf{\omega}(\mathbf{v}_f, \mathbf{v}_g) \\ & = \omega(v_f, v_g) \end{aligned} \,. \end{displaymath} Notice that on the right we have a \emph{constant} function on $V$. Write $\rho_2 : \mathbb{R} \hookrightarrow C^\infty(V)$ for the inclusion of the constant functions, and write \begin{displaymath} \rho_1 : V \stackrel{\omega}{\to} V^* \hookrightarrow C^\infty(V) \,. \end{displaymath} Then, by the above, the inclusion \begin{displaymath} \rho : V \oplus \mathbb{R} \stackrel{(\rho_1, \rho_2)}{\to} C^\infty(V) \end{displaymath} induces a Lie algebra [[homomorphism]] \begin{displaymath} \rho : Heis(V,\omega) \hookrightarrow \mathcal{P}(V, \mathbf{\omega}) \end{displaymath} which exhibits the Heisenberg Lie algebra as a [[subobject|sub]]-Lie algebra of that underlying the Poisson algebra. \end{proof} \hypertarget{integration_to_heisenberg_group}{}\subsubsection*{{Integration to Heisenberg group}}\label{integration_to_heisenberg_group} As for any [[Lie algebra]] one has [[Lie integration]] of the Heisenberg Lie algebra to a [[Lie group]]. This is called the \emph{[[Heisenberg group]]} (of the given [[symplectic vector space]]). \hypertarget{relation_to_the_weyl_algebra}{}\subsubsection*{{Relation to the Weyl algebra}}\label{relation_to_the_weyl_algebra} In the case of standard [[symplectic form]] on the [[Cartesian space]] $\mathbf{R}^{2n}$, the [[universal enveloping algebra]] of the Heisenberg Lie algebra is an [[associative algebra]] $\mathcal{U}(Heis(\mathbb{R}^{2n}))$. The [[quotient]] of this that identifies the [[center|central]] elements of the Heisenberg Lie algebra with multiples of the identity element is the \emph{[[Weyl algebra]]} on $n$ generators. \hypertarget{relation_to_the_heisenberg_double}{}\subsubsection*{{Relation to the Heisenberg double}}\label{relation_to_the_heisenberg_double} Given any [[Hopf algebra]], one can define its [[Heisenberg double]], which generalized the Heisenberg-Weyl algebra, which corresponds to the case when the Hopf algebra is the polynomial algebra. \hypertarget{references}{}\subsection*{{References}}\label{references} Lecture notes on standard material include \begin{itemize}% \item (section 4 in) Gordon, \emph{Infinite-dimensional Lie algebras}, Lecture notes, Edinburgh (2008) (\href{http://www.maths.ed.ac.uk/~igordon/LA1.pdf}{pdf}) \item Teruji Thomas, \emph{Geometric quantization II: Prequantization and the Heisenberg group} (\href{http://www.maths.ed.ac.uk/~jthomas7/GeomQuant/Lecture2.pdf}{pdf}), section 4 (relating to [[geometric quantization]]) \end{itemize} A [[categorification]] of the Heisenberg algebra is considered in \begin{itemize}% \item [[Mikhail Khovanov]], \emph{Heisenberg algebra and a graphical calculus} (\href{http://arxiv.org/abs/1009.3295}{arXiv:1009.3295}) \item [[Owen Gwilliam]], [[Rune Haugseng]], \emph{Linear Batalin-Vilkovisky quantization as a functor of ∞-categories} (\href{https://arxiv.org/abs/1608.01290}{arXiv:1608.01290}) \end{itemize} An $n$-fold categorification of the Lie algebra underlying the [[Poisson algebra]] (and hence including the Weil algebra) for all $n$ to a [[Lie n-algebra]] is considered in [[n-plectic geometry]], \begin{itemize}% \item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068}) \end{itemize} [[!redirects Heisenberg Lie algebras]] [[!redirects Heisenberg algebra]] [[!redirects Heisenberg algebras]] \end{document}